11. But now by measuring the diameters of each of these rings, and

knowing the convexity of the telescope glass, the thickness of the

water at each of those rings may be determined with great exactness:

for instance the thickness of it, where the white light of the first

order is reflected, is about 3⅞ such parts, of which an inch contains

1000000[303]. And this measure gives the thickness of the bubble, where

it appeared of this white colour, as well as of the water between the

glasses; though the transparent body which surrounds the water in these

two cases be very different: for our author found, that the condition

of the ambient body would not alter the species of the colour at all,

though it might its strength and brightness; for pieces of Muscovy

glass, which were so thin as to appear coloured by being wet with

water, would have their colours faded and made less bright thereby; but

he could not observe their species at all to be changed. So that the

thickness of any transparent body determines its colour, whatever body

the light passes through in coming to it[304].

12. BUT it was found that different transparent bodies would not under

the same thicknesses exhibit the same colours: for if the forementioned

glasses were laid upon each other without any water between their

surfaces, the air itself would afford the same colours as the water,

but more expanded, insomuch that each ring had a larger diameter, and

all in the same proportion. So that the thickness of the air proper to

each colour was in the same proportion larger, than the thickness of

the water appropriated to the same[305].

13. IF we examine with care all the circumstances of these colours,

which will be enumerated in the next chapter, we shall not be

surprized, that our author takes them to bear a great analogy to the

colours of natural bodies[306]. For the regularity of those various and

strange appearances relating to them, which makes the most mysterious

part of the action between light and bodies, as the next chapter will

shew, is sufficient to convince us that the principle, from which

they flow, is of the greatest importance in the frame of nature; and

therefore without question is designed for no less a purpose than to

give bodies their various colours, to which end it seems very fitly

suited. For if any such transparent substance of the thickness

proper to produce any one colour should be cut into slender threads,

or broken into fragments, it does not appear but these should retain

the same colour; and a heap of such fragments should frame a body of

that colour. So that this is without dispute the cause why bodies are

of this or the other colour, that the particles of which they are

composed are of different sizes. Which is farther confirmed by the

analogy between the colours of thin plates, and the colours of many

bodies. For example, these plates do not look of the same colour when

viewed obliquely, as when seen direct; for if the rings and colours

between a convex and plane glass are viewed first in a direct manner,

and then at different degrees of obliquity, the rings will be observed

to dilate themselves more and more as the obliquity is increased[307];

which shews that the transparent substance between the glasses does

not exhibit the same colour at the same thickness in all situations of

the eye: just so the colours in the very same part of a peacock’s tail

change, as the tail changes posture in respect of the sight. Also the

colours of silks, cloths, and other substances, which water or oyl can

intimately penetrate, become faint and dull by the bodies being wet

with such fluids, and recover their brightness again when dry; just

as it was before said that plates of Muscovy glass grew faint and dim

by wetting. To this may be added, that the colours which painters use

will be a little changed by being ground very elaborately, without

question by the diminution of their parts. All which particulars, and

many more that might be extracted from our author, give abundant

proof of the present point. I shall only subjoin one more: these

transparent plates transmit through them all the light they do not

reflect; so that when looked through they exhibit those colours, which

result from the depriving white light of the colour reflected. This

may commodiously be tryed by the glasses so often mentioned; which if

looked through exhibit coloured rings as by reflected light, but in a

contrary order; for the middle spot, which in the other view appears

black for want of reflected light, now looks perfectly white, opposite

to the blue circle; next without this spot the light appears tinged

with a yellowish red; where the white circle appeared before, it now

seems dark; and so of the rest[308]. Now in the same manner, the light

transmitted through foliated gold into a darkened room appears greenish

by the loss of the yellow light, which gold reflects.

14. HENCE it follows, that the colours of bodies give a very probable

ground for making conjecture concerning the magnitude of their

constituent particles[309]. My reason for calling it a conjecture is,

its being difficult to fix certainly the order of any colour. The

green of vegetables our author judges to be of the third order, partly

because of the intenseness of their colour; and partly from the changes

they suffer when they wither, turning at first into a greenish or more

perfect yellow, and afterwards some of them to an orange or red; which

changes seem to be effected from their ringing particles growing denser

by the exhalation of their moisture, and perhaps augmented likewise

by the accretion of the earthy and oily parts of that moisture. How

the mentioned colours should arise from increasing the bulk of those

particles, is evident; seeing those colours lie without the ring

of green between the glasses, and are therefore formed where the

transparent substance which reflects them is thicker. And that the

augmentation of the density of the colorific particles will conspire

to the production of the same effect, will be evident; if we remember

what was said of the different size of the rings, when air was included

between the glasses, from their size when water was between them; which

shewed that a substance of a greater density than another gives the

same colour at a less thickness. Now the changes likely to be wrought

in the density or magnitude of the parts of vegetables by withering

seem not greater, than are sufficient to change their colour into those

of the same order; but the yellow and red of the fourth order are not

full enough to agree with those, into which these substances change,

nor is the green of the second sufficiently good to be the colour of

vegetables; so that their colour must of necessity be of the third

order.

15. THE blue colour of syrup of violets our author supposes to be of

the third order; for acids, as vinegar, with this syrup change it red,

and salt of tartar or other alcalies mixed therewith turn it green.

But if the blue colour of the syrup were of the second order, the red

colour, which acids by attenuating its parts give it, must be of the

first order, and the green given it by alcalies by incrassating its

particles should be of the second; whereas neither of those colours is

perfect enough, especially the green, to answer those produced by these

changes; but the red may well enough be allowed to be of the second

order, and the green of the third; in which case the blue must be

likewise of the third order.

16. THE azure colour of the skies our author takes to be of the

first order, which requires the smallest particles of any colour,

and therefore most like to be exhibited by vapours, before they have

sufficiently coalesced to produce clouds of other colours.

17. THE most intense and luminous white is of the first order, if

less strong it is a mixture of the colours of all the orders. Of

the latter sort he takes the colour of linnen, paper, and such like

substances to be; but white metals to be of the former sort. The

arguments for it are these. The opacity of all bodies has been shewn

to arise from the number and strength of the reflections made within

them; but all experiments shew, that the strongest reflection is made

at those surfaces, which intercede transparent bodies differing most

in density. Among other instances of this, the experiments before us

afford one; for when air only is included between the glasses, the

coloured rings are not only more dilated, as has before been said,

than when water is between them; but are likewise much more luminous

and bright. It follows therefore, that whatever medium pervades the

pores of bodies, if so be there is any, those substances must be most

opake, the density of whose parts differs most from the density of the

medium, which fills their pores. But it has been sufficiently proved

in the former part of this tract, that there is no very dense medium

lodging in, at least pervading at liberty the pores of bodies. And it

is farther proved by the present experiments. For when air is inclosed

by the denser substance of glass, the rings dilate themselves, as

has been said, by being viewed obliquely; this they do so very much,

that at different obliquities the same thickness of air will exhibit

all sorts of colours. The bubble of water, though surrounded with the

thinner substance of air, does likewise change its colour by being

viewed obliquely; but not any thing near so much, as in the other case;

for in that the same colour might be seen, when the rings were viewed

most obliquely, at more than twelve times the thickness it appeared

at under a direct view; whereas in this other case the thickness was

never found considerably above half as much again. Now the colours of

bodies not depending only on the light, that is incident upon them

perpendicularly, but likewise upon that, which falls on them in all

degrees of obliquity; if the medium surrounding their particles were

denser than those particles, all sorts of colours must of necessity

be reflected from them so copiously, as would make the colours of all

bodies white, or grey, or at best very dilute and imperfect. But on

the other hand, if the medium in the pores of bodies be much rarer

than their particles, the colour reflected will be so little changed

by the obliquity of the rays, that the colour produced by the rays,

which fall near the perpendicular, may so much abound in the reflected

light, as to give the body their colour with little allay. To this

may be added, that when the difference of the contiguous transparent

substances is the same, a colour reflected from the denser substance

reduced into a thin plate and surrounded by the rarer will be more

brisk, than the same colour will be, when reflected from a thin plate

formed of the rarer substance, and surrounded by the denser; as our

author experienced by blowing glass very thin at a lamp furnace,

which exhibited in the open air more vivid colours, than the air does

between two glasses. From these considerations it is manifest, that

if all other circumstances are alike, the densest bodies will be most

opake. But it was observed before, that these white metals can hardly

be made so thin, except by being dissolved in corroding liquors, as

to be rendred transparent; though none of them are so dense as gold,

which proves their great opacity to have some other cause besides their

density; and none is more fit to produce this, than such a size of

their particles, as qualifies them to reflect the white of the first

order.

18. FOR producing black the particles ought to be smaller than for

exhibiting any of the colours, viz. of a size answering to the

thickness of the bubble, where by reflecting little or no light it

appears colourless; but yet they must not be too small, for that will

make them transparent through deficiency of reflections in the inward

parts of the body, sufficient to stop the light from going through it;

but they must be of a size bordering upon that disposed to reflect

the faint blue of the first order, which affords an evident reason why

blacks usually partake a little of that colour. We see too, why bodies

dissolved by fire or putrefaction turn black: and why in grinding

glasses upon copper plates the dust of the glass, copper, and sand it

is ground with, become very black: and in the last place why these

black substances communicate so easily to others their hue; which is,

that their particles by reason of the great minuteness of them easily

overspread the grosser particles of others.

19. I SHALL now finish this chapter with one remark of the exceeding

great porosity in bodies necessarily required in all that has here

been said; which, when duly considered, must appear very surprizing;

but perhaps it will be matter of greater surprize, when I affirm that

the sagacity of our author has discovered a method, by which bodies

may easily become so; nay how any the least portion of matter may be

wrought into a body of any assigned dimensions how great so ever, and

yet the pores of that body none of them greater, than any the smallest

magnitude proposed at pleasure; notwithstanding which the parts of the

body shall so touch, that the body itself shall be hard and solid[310].

The manner is this: suppose the body be compounded of particles of

such figures, that when laid together the pores found between them

may be equal in bigness to the particles; how this may be effected,

and yet the body be hard and solid, is not difficult to understand;

and the pores of such a body may be made of any proposed degree of

smallness. But the solid matter of a body so framed will take up only

half the space occupied by the body; and if each constituent particle

be composed of other less particles according to the same rule, the

solid parts of such a body will be but a fourth part of its bulk; if

every one of these lesser particles again be compounded in the same

manner, the solid parts of the whole body shall be but one eighth of

its bulk; and thus by continuing the composition the solid parts of the

body may be made to bear as small a proportion to the whole magnitude

of the body, as shall be desired, notwithstanding the body will be by

the contiguity of its parts capable of being in any degree hard. Which

shews that this whole globe of earth, nay all the known bodies in the

universe together, as far as we know, may be compounded of no greater

a portion of solid matter, than might be reduced into a globe of one

inch only in diameter, or even less. We see therefore how by this means

bodies may easily be made rare enough to transmit light, with all

that freedom pellucid bodies are found to do. Though what is the real

structure of bodies we yet know not.

~CHAP. III.~

Of the REFRACTION, REFLECTION, and INFLECTION of LIGHT.

THUS much of the colours of natural bodies; our method now leads us

to speculations yet greater, no less than to lay open the causes of

all that has hitherto been related. For it must in this chapter be

explained, how the prism separates the colours of the sun’s light, as

we found in the first chapter; and why the thin transparent plates

discoursed of in the last chapter, and consequently the particles of

coloured bodies, reflect that diversity of colours only by being of

different thicknesses.

2. FOR the first it is proved by our author, that the colours of the

sun’s light are manifested by the prism, from the rays undergoing

different degrees of refraction; that the violet-making rays, which

go to the upper part of the coloured image in the first experiment

of the first chapter, are the most refracted; that the indigo-making

rays are refracted, or turned out of their course by passing through

the prism, something less than the violet-making rays, but more than

the blue-making rays; and the blue-making rays more than the green;

the green-making rays more than the yellow; the yellow more than the

orange; and the orange-making rays more than the red-making, which are

least of all refracted. The first proof of this, that rays of different

colours are refracted unequally is this. If you take any body, and

paint one half of it red and the other half blue, then upon viewing

it through a prism those two parts shall appear separated from each

other; which can be caused no otherwise than by the prism’s refracting

the light of one half more than the light of the other half. But the

blue half will be most refracted; for if the body be seen through the

prism in such a situation, that the body shall appear lifted upwards

by the refraction, as a body within a bason of water, in the experiment

mentioned in the first chapter, appeared to be lifted up by the

refraction of the water, so as to be seen at a greater distance than

when the bason is empty, then shall the blue part appear higher than

the red; but if the refraction of the prism be the contrary way, the

blue part shall be depressed more than the other. Again, after laying

fine threads of black silk across each of the colours, and the body

well inlightened, if the rays coming from it be received upon a convex

glass, so that it may by refracting the rays cast the image of the body

upon a piece of white paper held beyond the glass; then it will be seen

that the black threads upon the red part of the image, and those upon

the blue part, do not at the same time appear distinctly in the image

of the body projected by the glass; but if the paper be held so, that

the threads on the blue part may distinctly appear, the threads cannot

be seen distinct upon the red part; but the paper must be drawn farther

off from the convex glass to make the threads on this part visible; and

when the distance is great enough for the threads to be seen in this

red part, they become indistinct in the other. Whence it appears that

the rays proceeding from each point of the blue part of the body are

sooner united again by the convex glass than the rays which come from

each point of the red parts[311]. But both these experiments prove that

the blue-making rays, as well in the small refraction of the convex

glass, as in the greater refraction of the prism, are more bent, than

the red-making rays.

3. THIS seems already to explain the reason of the coloured spectrum

made by refracting the sun’s light with a prism, though our author

proceeds to examine that in particular, and proves that the different

coloured rays in that spectrum are in different degrees refracted; by

shewing how to place the prism in such a posture, that if all the rays

were refracted in the same manner, the spectrum should of necessity

be round: whereas in that case if the angle made by the two surfaces

of the prism, through which the light passes, that is the angle D F

E in fig. 126, be about 63 or 64 degrees, the image instead of being

round shall be near five times as long as broad; a difference enough

to shew a great inequality in the refractions of the rays, which

go to the opposite extremities of the image. To leave no scruple

unremoved, our author is very particular in shewing by a great number

of experiments, that this inequality of refraction is not casual, and

that it does not depend upon any irregularities of the glass; no nor

that the rays are in their passage through the prism each split and

divided; but on the contrary that every ray of the sun has its own

peculiar degree of refraction proper to it, according to which it is

more or less refracted in passing through pellucid substances always

in the same manner[312]. That the rays are not split and multiplied

by the refraction of the prism, the third of the experiments related

in our first chapter shews very clearly; for if they were, and the

length of the spectrum in the first refraction were thereby occasioned,

the breadth should be no less dilated by the cross refraction of the

second prism; whereas the breadth is not at all increased, but the

image is only thrown into an oblique posture by the upper part of the

rays which were at first more refracted than the under part, being

again turned farthest out of their course. But the experiment most

expressly adapted to prove this regular diversity of refraction is

this, which follows[313]. Two boards A B, C D (in fig. 130.) being

erected in a darkened room at a proper distance, one of them A B being

near the window-shutter E F, a space only being left for the prism G

H I to be placed between them; so that the rays entring at the hole M

of the window-shutter may after passing through the prism be trajected

through a smaller hole K made in the board A B, and passing on from

thence go out at another hole L made in the board C D of the same size

as the hole K, and small enough to transmit the rays of one colour

only at a time; let another prism N O P be placed after the board C

D to receive the rays passing through the holes K and L, and after

refraction by that prism let those rays fall upon the white surface

Q R. Suppose first the violet light to pass through the holes, and

to be refracted by the prism N O P to _s_, which if the prism N O P

were removed should have passed right onto W. If the prism G H I be

turned slowly about, while the boards and prism N O P remain fixed,

in a little time another colour will fall upon the hole L, which, if

the prism N O P were taken away, would proceed like the former rays

to the same point W; but the refraction of the prism N O P shall not

carry these rays to _s_, but to some place less distant from W as to

_t_. Suppose now the rays which go to _t_ to be the indigo-making rays.

It is manifest that the boards A B, C D, and prism N O P remaining

immoveable, both the violet-making and indigo-making rays are incident

alike upon the prism N O P, for they are equally inclined to its

surface O P, and enter it in the same part of that surface; which shews

that the indigo-making rays are less diverted out of their course by

the refraction of the prism, than the violet-making rays under an

exact parity of all circumstances. Farther, if the prism G H I be more

turned about, ’till the blue-making rays pass through the hole L, these

shall fall upon the surface Q R below I, as at _v_, and therefore are

subjected to a less refraction than the indigo-making rays. And thus

by proceeding it will be found that the green-making rays are less

refracted than the blue-making rays, and so of the rest, according to

the order in which they lie in the coloured spectrum.

4. THIS disposition of the different coloured rays to be refracted

some more than others our author calls their respective degrees of

refrangibility. And since this difference of refrangibility discovers

it self to be so regular, the next step is to find the rule it observes.

5. IT is a common principle in optics, that the sine of the angle of

incidence bears to the sine of the refracted angle a given proportion.

If A B (in fig. 131, 132) represent the surface of any refracting

substance, suppose of water or glass, and C D a ray of light incident

upon that face in the point D, let D E be the ray, after it has passed

the surface A B; if the ray pass out of the air into the substance

whose surface is A B (as in fig. 131) it shall be turned from the

surface, and if it pass out of that substance into air it shall be

bent towards it (as in fig. 132) But if F G be drawn through the point

D perpendicular to the surface A B, the angle under C D F made by the

incident ray and this perpendicular is called the angle of incidence;

and the angle under E D G, made by this perpendicular and the ray after

refraction, is called the refracted angle. And if the circle H F I G

be described with any interval cutting C D in H and D E in I, then

the perpendiculars H K, I L being let fall upon F G, H K is called

the sine of the angle under C D F the angle of incidence, and I L the

sine of the angle under E D G the refracted angle. The first of these

sines is called the sine of the angle of incidence, or more briefly the

sine of incidence, the latter is the sine of the refracted angle, or

the sine of refraction. And it has been found by numerous experiments

that whatever proportion the sine of incidence H K bears to the sine

of refraction I L in any one case, the same proportion shall hold in

all cases; that is, the proportion between these sines will remain

unalterably the same in the same refracting substance, whatever be the

magnitude of the angle under C D F.

6. BUT now because optical writers did not observe that every beam of

white light was divided by refraction, as has been here explained,

this rule collected by them can only be understood in the gross of the

whole beam after refraction, and not so much of any particular part

of it, or at most only of the middle part of the beam. It therefore

was incumbent upon our author to find by what law the rays were parted

from each other; whether each ray apart obtained this property, and

that the separation was made by the proportion between the sines of

incidence and refraction being in each species of rays different; or

whether the light was divided by some other rule. But he proves by a

certain experiment that each ray has its sine of incidence proportional

to its sine of refraction; and farther shews by mathematical reasoning,

that it must be so upon condition only that bodies refract the light

by acting upon it, in a direction perpendicular to the surface of the

refracting body, and upon the same sort of rays always in an equal

degree at the same distances[314].

7. OUR great author teaches in the next place how from the refraction

of the most refrangible and least refrangible rays to find the

refraction of all the intermediate ones[315]. The method is this:

if the sine of incidence be to the sine of refraction in the least

refrangible rays as A to B C, (in fig. 133) and to the sine of

refraction in the most refrangible as A to B D; if C E be taken equal

to C D, and then E D be so divided in F, G, H, I, K, L, that E D, E

F, E G, E H, E I, E K, E L, E C, shall be proportional to the eight

lengths of musical chords, which found the notes in an octave, E D

being the length of the key, E F the length of the tone above that

key, E G the length of the lesser third, E H of the fourth, E I of the

fifth, E K of the greater sixth, E L of the seventh, and E C of the

octave above that key; that is if the lines E D, E F, E G, E H, E I, E

K, E L, and E C bear the same proportion as the numbers, 1, 9/8, 5/6,

¾, ⅓, ¾, 9/61, ½, respectively then shall B D, B F, be the two limits

of the sines of refraction of the violet-making rays, that is the

violet-making rays shall not all of them have precisely the same sine

of refraction, but none of them shall have a greater sine than B D,

nor a less than B F, though there are violet-making rays which answer

to any sine of refraction that can be taken between these two. In the

same manner B F and B G are the limits of the sines of refraction

of the indigo-making rays; B G, B H are the limits belonging to the

blue-making rays; B H, B I the limits pertaining to the green-making

rays, B I, B K the limits for the yellow-making rays; B K, B L the

limits for the orange-making rays; and lastly, B L and B C the extreme

limits of the sines of refraction belonging to the red-making rays.

These are the proportions by which the heterogeneous rays of light are

separated from each other in refraction.

8. WHEN light passes out of glass into air, our author found A to B C

as 50 to 77, and the same A to B D as 50 to 78. And when it goes out

of any other refracting substance into air, the excess of the sine

of refraction of any one species of rays above its sine of incidence

bears a constant proportion, which holds the same in each species, to

the excess of the sine of refraction of the same sort of rays above

the sine of incidence into the air out of glass; provided the sines

of incidence both in glass and the other substance are equal. This

our author verified by transmitting the light through prisms of glass

included within a prismatic vessel of water; and draws from those

experiments the following observations: that whenever the light in

passing through so many surfaces parting diverse transparent substances

is by contrary refractions made to emerge into the air in a direction

parallel to that of its incidence, it will appear afterwards white at

any distance from the prisms, where you shall please to examine it;

but if the direction of its emergence be oblique to its incidence, in

receding from the place of emergence its edges shall appear tinged with

colours: which proves that in the first case there is no inequality

in the refractions of each species of rays, but that when any one

species is so refracted as to emerge parallel to the incident rays,

every sort of rays after refraction shall likewise be parallel to the

same incident rays, and to each other; whereas on the contrary, if the

rays of any one sort are oblique to the incident light, the several

species shall be oblique to each other, and be gradually separated by

that obliquity. From hence he deduces both the forementioned theorem,

and also this other; that in each sort of rays the proportion of the

sine of incidence to the sine of refraction, in the passage of the ray

out of any refracting substance into another, is compounded of the

proportion to which the sine of incidence would have to the sine of

refraction in the passage of that ray out of the first substance into

any third, and of the proportion which the sine of incidence would

have to the sine of refraction in the passage of the ray out of that

third substance into the second. From so simple and plain an experiment

has our most judicious author deduced these important theorems, by

which we may learn how very exact and circumspect he has been in this

whole work of his optics; that notwithstanding his great particularity

in explaining his doctrine, and the numerous collection of experiments

he has made to clear up every doubt which could arise, yet at the same

time he has used the greatest caution to make out every thing by the

simplest and easiest means possible.

9. OUR author adds but one remark more upon refraction, which is, that

if refraction be performed in the manner he has supposed from the

light’s being pressed by the refracting power perpendicularly toward

the surface of the refracting body, and consequently be made to move

swifter in the body than before its incidence; whether this power act

equally at all distances or otherwise, provided only its power in the

same body at the same distances remain without variation the same in

one inclination of the incident rays as well as another; he observes

that the refracting powers in different bodies will be in the duplicate

proportion of the tangents of the lead angles, which the refracted

light can make with the surfaces of the refracting bodies[316].

This observation may be explained thus. When the light passes into

any refracting substance, it has been shewn above that the sine of

incidence bears a constant proportion to the sine of refraction.

Suppose the light to pass to the refracting body A B C D (in fig.

134) in the line E F, and to fall upon it at the point F, and then to

proceed within the body in the line F G. Let H I be drawn through F

perpendicular to the surface A B, and any circle K L M N be described

to the center F. Then from the points O and P where this circle cuts

the incident and refracted ray, the perpendiculars O Q, P R being

drawn, the proportion of O Q to P R will remain the same in all the

different obliquities, in which the same ray of light can fall on the

surface A B. Now O Q is less than F L the semidiameter of the circle K

L M N, but the more the ray E F is inclined down toward the surface A

B, the greater will O Q be, and will approach nearer to the magnitude

of F L. But the proportion of O Q to P R remaining always the same,

when O Q, is largest, P R will also be greatest; so that the more the

incident ray E F is inclined toward the surface A B, the more the ray

F G after refraction will be inclined toward the same. Now if the line

F S T be so drawn, that S V being perpendicular to F I shall be to F L

the semidiameter of the circle in the constant proportion of P R to O

Q; then the angle under N F T is that which I meant by the least of all

that can be made by the refracted ray with this surface, for the ray

after refraction would proceed in this line, if it were to come to the

point F lying on the very surface A B; for if the incident ray came to

the point F in any line between A F and F H, the ray after refraction

would proceed forward in some line between F T and F I. Here if N W

be drawn perpendicular to F N, this line N W in the circle K L M N is

called the tangent of the angle under N F S. Thus much being premised,

the sense of the forementioned proposition is this. Let there be two

refracting substances (in fig. 135) A B C D, and E F G H. Take a point,

as I, in the surface A B, and to the center I with any semidiameter

describe the circle K L M. In like manner on the surface E F take

some point N, as a center, and describe with the same semidiameter

the circle O P Q. Let the angle under B I R be the least which the

refracted light can make with the surface A B, and the angle under F N

S the least which the refracted light can make with the surface E F.

Then if L T be drawn perpendicular to A B, and P V perpendicular to E

F; the whole power, wherewith the substance A B C D acts on the light,

will bear to the whole power wherewith the substance E F G H acts on,

the light, a proportion, which is duplicate of the proportion, which L

T bears to P V.

10. UPON comparing according to this rule the refractive powers of a

great many bodies it is found, that unctuous bodies which abound most

with sulphureous parts refract the light two or three times more in

proportion to their density than others: but that those bodies, which

seem to receive in their composition like proportions of sulphureous

parts, have their refractive powers proportional to their densities; as

appears beyond contradiction by comparing the refractive power of so

rare a substance as the air with that of common glass or rock crystal,

though these substances are 2000 times denser than air; nay the same

proportion is found to hold without sensible difference in comparing

air with pseudo-topar and glass of antimony, though the pseudo-topar

be 3500 times denser than air, and glass of antimony no less than

4400 times denser. This power in other substances, as salts, common

water, spirit of wine, &c. seems to bear a greater proportion to their

densities than these last named, according as they abound with sulphurs

more than these; which makes our author conclude it probable, that

bodies act upon the light chiefly, if not altogether, by means of the

sulphurs in them; which kind of substances it is likely enters in some

degree the composition of all bodies. Of all the substances examined

by our author, none has so great a refractive power, in respect of its

density, as a diamond.

11. OUR author finishes these remarks, and all he offers relating to

refraction, with observing, that the action between light and bodies is

mutual, since sulphureous bodies, which are most readily set on fire

by the sun’s light, when collected upon them with a burning glass, act

more upon light in refracting it, than other bodies of the same density

do. And farther, that the densest bodies, which have been now shewn to

act most upon light, contract the greatest heat by being exposed to the

summer sun.

12. HAVING thus dispatched what relates to refraction, we must address

ourselves to discourse of the other operation of bodies upon light

in reflecting it. When light passes through a surface, which divides

two transparent bodies differing in density, part of it only is

transmitted, another part being reflected. And if the light pass out

of the denser body into the rarer, by being much inclined to the

foresaid surface at length no part of it shall pass through, but be

totally reflected. Now that part of the light, which suffers the

greatest refraction, shall be wholly reflected with a less obliquity

of the rays, than the parts of the light which undergo a less degree

of refraction; as is evident from the last experiment recited in the

first chapter; where, as the prisms D E F, G H I, (in fig. 129.) were

turned about, the violet light was first totally reflected, and then

the blue, next to that the green, and so of the rest. In consequence

of which our author lays down this proportion; that the sun’s light

differs in reflexibility, those rays being most reflexible, which are

most refrangible. And collects from this, in conjunction with other

arguments, that the refraction and reflection, of light are produced

by the same cause, compassing those different effects only by the

difference of circumstances with which it is attended. Another proof

of this being taken by our author from what he has discovered of

the passage of light through thin transparent plates, viz. that any

particular species of light, suppose, for instance, the red-making

rays, will enter and pass out of such a plate, if that plate be of

some certain thicknesses; but if it be of other thicknesses, it will

not break through it, but be reflected back: in which is seen, that

the thickness of the plate determines whether the power, by which that

plate acts upon the light, shall reflect it, or suffer it to pass

through.

13. BUT this last mentioned surprising property of the action between

light and bodies affords the reason of all that has been said in the

preceding chapter concerning the colours of natural bodies; and must

therefore more particularly be illustrated and explained, as being what

will principally unfold the nature of the action of bodies upon light.

14. TO begin: The object glass of a long telescope being laid upon a

plane glass, as proposed in the foregoing chapter, in open day-light

there will be exhibited rings of various colours, as was there related;

but if in a darkened room the coloured spectrum be formed by the prism,

as in the first experiment of the first chapter, and the glasses be

illuminated by a reflection from the spectrum, the rings shall not

in this case exhibit the diversity of colours before described, but

appear all of the colour of the light which falls upon the glasses,

having dark rings between. Which shews that the thin plate of air

between the glasses at some thicknesses reflects the incident light,

at other places does not reflect it, but is found in those places to

give the light passage; for by holding the glasses in the light as

it passes from the prism to the spectrum, suppose at such a distance

from the prism that the several sorts of light must be sufficiently

separated from each other, when any particular sort of light falls

on the glasses, you will find by holding a piece of white paper at a

small distance beyond the glasses, that at those intervals, where the

dark lines appeared upon the glasses, the light is so transmitted,

as to paint upon the paper rings of light having that colour which

falls upon the glasses. This experiment therefore opens to us this

very strange property of reflection, that in these thin plates it

should bear such a relation to the thickness of the plate, as is here

shewn. Farther, by carefully measuring the diameters of each ring it

is found, that whereas the glasses touch where the dark spot appears

in the center of the rings made by reflexion, where the air is of

twice the thickness at which the light of the first ring is reflected,

there the light by being again transmitted makes the first dark ring;

where the plate has three times that thickness which exhibits the

first lucid ring, it again reflects the light forming the second lucid

ring; when the thickness is four times the first, the light is again

transmitted so as to make the second dark ring; where the air is five

times the first thickness, the third lucid ring is made; where it has

six times the thickness, the third dark ring appears, and so on: in

so much that the thicknesses, at which the light is reflected, are

in proportion to the numbers 1, 3, 5, 7, 9, &c. and the thicknesses,

where the light is transmitted, are in the proportion of the numbers

0, 2, 4, 6, 8, &c. And these proportions between the thicknesses which

reflect and transmit the light remain the same in all situations of the

eye, as well when the rings are viewed obliquely, as when looked on

perpendicularly. We must farther here observe, that the light, when it

is reflected, as well as when it is transmitted, enters the thin plate,

and is reflected from its farther surface; because, as was before

remarked, the altering the transparent body behind the farther surface

alters the degree of reflection as when a thin piece of Muscovy glass

has its farther surface wet with water, and the colour of the glass

made dimmer by being so wet; which shews that the light reaches to the

water, otherwise its reflection could not be influenced by it. But

yet this reflection depends upon some power propagated from the first

surface to the second; for though made at the second surface it depends

also upon the first, because it depends upon the distance between the

surfaces; and besides, the body through which the light passes to the

first surface influences the reflection: for in a plate of Muscovy

glass, wetting the surface, which first receives the light, diminishes

the reflection, though not quite so much as wetting the farther surface

will do. Since therefore the light in passing through these thin plates

at some thicknesses is reflected, but at others transmitted without

reflection, it is evident, that this reflection is caused by some

power propagated from the first surface, which intermits and returns

successively. Thus is every ray apart disposed to alternate reflections

and transmissions at equal intervals; the successive returns of which

disposition our author calls the fits of easy reflection, and of easy

transmission. But these fits, which observe the same law of returning

at equal intervals, whether the plates are viewed perpendicularly or

obliquely, in different situations of the eye change their magnitude.

For what was observed before in respect of those rings, which appear

in open day-light, holds likewise in these rings exhibited by simple

lights; namely, that these two alter in bigness according to the

different angle under which they are seen: and our author lays down a

rule whereby to determine the thicknesses of the plate of air, which

shall exhibit the same colour under different oblique views[317]. And

the thickness of the aereal plate, which in different inclinations of

the rays will exhibit to the eye in open day-light the same colour, is

also varied by the same rule[318]. He contrived farther a method of

comparing in the bubble of water the proportion between the thickness

of its coat, which exhibited any colour when seen perpendicularly,

to the thickness of it, where the same colour appeared by an oblique

view; and he found the same rule to obtain here likewise[319]. But

farther, if the glasses be enlightened successively by all the several

species of light, the rings will appear of different magnitudes; in

the red light they will be larger than in the orange colour, in that

larger than in the yellow, in the yellow larger than in the green,

less in the blue, less yet in the indigo, and least of all in the

violet: which shew that the same thickness of the aereal plate is not

fitted to reflect all colours, but that one colour is reflected where

another would have been transmitted; and as the rays which are most

strongly refracted form the least rings, a rule is laid down by our

author for determining the relation, which the degree of refraction of

each species of colour has to the thicknesses of the plate where it is

reflected.

15. FROM these observations our author shews the reason of that great

variety of colours, which appears in these thin plates in the open

white light of the day. For when this white light falls on the plate,

each part of the light forms rings of its own colour; and the rings

of the different colours not being of the same bigness are variously

intermixed, and form a great variety of tints[320].

16. IN certain experiments, which our author made with thick glasses,

he found, that these fits of easy reflection and transmission returned

for some thousands of times, and thereby farther confirmed his

reasoning concerning them[321].

17. UPON the whole, our great author concludes from some of the

experiments made by him, that the reason why all transparent bodies

refract part of the light incident upon them, and reflect another

part, is, because some of the light, when it comes to the surface of

the body, is in a fit of easy transmission, and some part of it in

a fit of easy reflection; and from the durableness of these fits he

thinks it probable, that the light is put into these fits from their

first emission out of the luminous body; and that these fits continue

to return at equal intervals without end, unless those intervals be

changed by the light’s entring into some refracting substance[322]. He

likewise has taught how to determine the change which is made of the

intervals of the fits of easy transmission and reflection, when the

light passes out of one transparent space or substance into another.

His rule is, that when the light passes perpendicularly to the surface,

which parts any two transparent substances, these intervals in the

substance, out of which the light passes, bear to the intervals in the

substance, whereinto the light enters, the same proportion, as the sine

of incidence bears to the sine of refraction[323]. It is farther to be

observed, that though the fits of easy reflection return at constant

intervals, yet the reflecting power never operates, but at or near a

surface where the light would suffer refraction; and if the thickness

of any transparent body shall be less than the intervals of the fits,

those intervals shall scarce be disturbed by such a body, but the light

shall pass through without any reflection[324].

18. WHAT the power in nature is, whereby this action between light and

bodies is caused, our author has not discovered. But the effects, which

he has discovered, of this power are very surprising, and altogether

wide from any conjectures that had ever been framed concerning it; and

from these discoveries of his no doubt this power is to be deduced,

if we ever can come to the knowledge of it. Sir ISAAC NEWTON has in

general hinted at his opinion concerning it; that probably it is

owing to some very subtle and elastic substance diffused through the

universe, in which such vibrations may be excited by the rays of

light, as they pass through it, that shall occasion it to operate so

differently upon the light in different places as to give rise to these

alternate fits of reflection and transmission, of which we have now

been speaking[325]. He is of opinion, that such a substance may produce

this and other effects also in nature, though it be so rare as not to

give any sensible resistance to bodies in motion[326]; and therefore

not inconsistent with what has been said above, that the planets move

in spaces free from resistance[327].

19. IN order for the more full discovery of this action between light

and bodies, our author began another set of experiments, wherein he

found the light to be acted on as it passes near the edges of solid

bodies; in particular all small bodies, such as the hairs of a man’s

head or the like, held in a very small beam of the sun’s light, cast

extremely broad shadows. And in one of these experiments the shadow

was 35 times the breadth of the body[328]. These shadows are also

observed to be bordered with colours[329]. This our author calls the

inflection of light; but as he informs us, that he was interrupted

from prosecuting these experiments to any length, I need not detain my

readers with a more particular account of them.

~CHAP. IV.~

Of OPTIC GLASSES.

SIR ~ISAAC NEWTON~ having deduced from his doctrine of light and

colours a surprising improvement of telescopes, of which I intend

here to give an account, I shall first premise something in general

concerning those instruments.

2. IT will be understood from what has been said above, that when light

falls upon the surface of glass obliquely, after its entrance into

the glass it is more inclined to the line drawn through the point of

incidence perpendicular to that surface, than before. Suppose a ray of

light issuing from the point A (in fig. 136) falls on a piece of glass

B C D E, whose surface B C, whereon the ray falls, is of a spherical

or globular figure, the center whereof is F. Let the ray proceed in

the line A G falling on the surface B C in the point G, and draw F G

H. Here the ray after its entrance into the glass will pass on in some

line, as G I, more inclined toward the line F G H that the line A G is

inclined thereto; for the line F G H is perpendicular to the surface B

C in the point G. By this means, if a number of rays proceeding from

any one point fall on a convex spherical surface of glass, they shall

be inflected (as is represented in fig. 137,) so as to be gathered

pretty close together about the line drawn through the center of the

glass from the point, whence the rays proceed; which line henceforward

we shall call the axis of the glass: or the point from whence the rays

proceed may be so near the glass, that the rays shall after entring the

glass still go on to spread themselves, but not so much as before; so

that if the rays were to be continued backward (as in fig. 138,) they

should gather together about the axis at a place more remote from the

glass, than the point is, whence they actually proceed. In these and

the following figures A denotes the point to which the rays are related

before refraction, B the point to which they are directed afterwards,

and C the center of the refracting surface. Here we may observe, that

it is possible to form the glass of such a figure, that all the rays

which proceed from one point shall after refraction be reduced again

exactly into one point on the axis of the glass. But in glasses of a

spherical form though this does not happen; yet the rays, which fall

within a moderate distance from the axis, will unite extremely near

together. If the light fall on a concave spherical surface, after

refraction it shall spread quicker than before (as in fig. 139,) unless

the rays proceed from a point between the center and the surface of the

glass. If we suppose the rays of light, which fall upon the glass, not

to proceed from any point, but to move so as to tend all to some point

in the axis of the glass beyond the surface; if the glass have a convex

surface, the rays shall unite about the axis sooner, than otherwise

they would do (as in fig. 140,) unless the point to which they tended

was between the surface and the center of that surface. But if the

surface be concave, they shall not meet so soon: nay perhaps converge.

(See fig. 141 and 142.)

5. FARTHER, because the light in passing out of glass into the air is

turned by the refraction farther off from the line drawn through the

point of incidence perpendicular to the refracting surface, than it was

before; the light which spreads from a point shall by parting through

a convex surface of glass into the air be made either to spread less

than before (as in fig. 143,) or to gather about the axis beyond the

glass (as in fig. 144.) But if the rays of light were proceeding to a

point in the axis of the glass, they should by the refraction be made

to unite sooner about that axis (as in fig. 145.) If the surface of

the glass be concave, rays which proceed from a point shall be made to

spread faster (as in fig 146,) but rays which are tending to a point in

the axis of the glass, shall be made to gather about the axis farther

from the glass (as in fig. 147) or even to diverge (as in fig. 148,)

unless the point, to which the rays are directed, lies between the

surface of the glass and its center.

4. THE rays, which spread themselves from a point, are called

diverging; and such as move toward a point, are called converging rays.

And the point in the axis of the glass, about which the rays gather

after refraction, is called the focus of those rays.

5. IF a glass be formed of two convex spherical surfaces (as in fig.

149,) where the glass AB is formed of the surfaces A C B and A D B, the

line drawn through the centers of the two surfaces, as the line E F, is

called the axis of the glass; and rays, which diverge from any point of

this axis, by the refraction of the glass will be caused to converge

toward some part of the axis, or at least to diverge as from a point

more remote from the glass, than that from whence they proceeded; for

the two surfaces both conspire to produce this effect upon the rays.

But converging rays will be caused by such a glass as this to converge

sooner. If a glass be formed of two concave surfaces, as the glass A

B (in fig. 150,) the line C D drawn through the centers, to which the

two surfaces are formed, is called the axis of the glass. Such a glass

shall cause diverging rays, which proceed from any point in the axis

of the glass, to diverge much more, as if they came from some place in

the axis of the glass nearer to it than the point, whence the rays

actually proceed. But converging rays will be made either to converge

less, or even to diverge.

[Illustration]

6. IN these glasses rays, which proceed from any point near the axis,

will be affected as it were in the same manner, as if they proceeded

from the very axis it self, and such as converge toward a point at a

small distance from the axis will suffer much the same effects from

the glass, as if they converged to some point in the very axis. By

this means any luminous body exposed to a convex glass may have an

image formed upon any white body held beyond the glass. This may be

easily tried with a common spectacle-glass. For if such a glass be held

between a candle and a piece of white paper, if the distances of the

candle, glass, and paper be properly adjusted, the image of the candle

will appear very distinctly upon the paper, but be seen inverted; the

reason whereof is this. Let A B (in fig. 151) be the glass, C D an

object placed cross the axis of the glass. Let the rays of light, which

issue from the point E, where the axis of the glass crosses the object,

be so refracted by the glass, as to meet again about the point F. The

rays, which diverge from the point C of the object, shall meet again

almost at the same distance from the glass, but on the other side of

the axis, as at G; for the rays at the glass cross the axis. In like

manner the rays, which proceed from the point D, will meet about H on

the other side of the axis. None of these rays, neither those which

proceed from the point E in the axis, nor those which issue from C or

D, will meet again exactly in one point; but yet in one place, as is

here supposed at F, G, and H, they will be crouded so close together,

as to make a distinct image of the object upon any body proper to

reflect it, which shall be held there.

7. IF the object be too near the glass for the rays to converge after

the refraction, the rays shall issue out of the glass, as if they

diverged from a point more distant from the glass, than that from

whence they really proceed (as in fig. 152,) where the rays coming from

the point E of the object, which lies on the axis of the glass A B,

issue out of the glass, as if they came from the point F more remote

from the glass than E; and the rays proceeding from the point C issue

out of the glass, as if they proceeded from the point G; likewise the

rays which issue from the point D emerge out of the glass, as if they

came from the point H. Here the point G is on the same side of the

axis, as the point C; and the point H on the same side, as the point

D. In this case to an eye placed beyond the glass the object should

appear, as if it were in the situation G F H.

8. IF the glass A B had been concave (as in, fig. 153,) to an eye

beyond the glass the object C D would appear in the situation G H,

nearer to the glass than really it is. Here also the object will not be

inverted; but the point G is on the same side the axe with the point C,

and H on the same side as D.

9. HENCE may be understood, why spectacles made with convex glasses

help the sight in old age: for the eye in that age becomes unfit to

see objects distinctly, except such as are remov’d to a very great

distance; whence all men, when they first stand in need of spectacles,

are observed to read at arm’s length, and to hold the object at a

greater distance, than they used to do before. But when an object is

removed at too great a distance from the sight, it cannot be seen

clearly, by reason that a less quantity of light from the object will

enter the eye, and the whole object will also appear smaller. Now by

help of a convex glass an object may be held near, and yet the rays of

light issuing from it will enter the eye, as if the object were farther

removed.

10. AFTER the same manner concave glasses assist such, as are short

sighted. For these require the object to be brought inconveniently near

to the eye, in order to their seeing it distinctly; but by such a glass

the object may be removed to a proper distance, and yet the rays of

light enter the eye, as if they came from a place much nearer.

11. WHENCE these defects of the sight arise, that in old age

objects cannot be seen distinct within a moderate distance, and in

short-sightedness not without being brought too near, will be easily

understood, when the manner of vision in general shall be explain’d;

which I shall now endeavour to do, in order to be better understood in

what follows. The eye is form’d, as is represented in fig. 154. It is

of a globular figure, the fore part whereof scarce more protuberant

than the rest is transparent. Underneath this transparent part is a

small collection of an humour in appearance like water, and it has also

the same refractive power as common water; this is called the aqueous

humour, and fills the space A B C D in the figure. Next beyond lies the

body D E F G; this is solid but transparent, it is composed with two

convex surfaces, the hinder surface E F G being more convex, than the

anterior E D G. Between the outer membrane A B C, and this body E D G

F is placed that membrane, which exhibits the colours, that are seen

round the sight of the eye; and the black spot, which is called the

sight or pupil, is a hole in this membrane, through which the light

enters, whereby we see. This membrane is fixed only by its outward

circuit, and has a muscular power, whereby it dilates the pupil in

a weak light, and contracts it in a strong one. The body D E F G is

called the crystalline humour, and has a greater refracting power than

water. Behind this the bulk of the eye is filled up with what is called

the vitreous humor, this has much the same refractive power with water.

At the bottom of the eye toward the inner side next the nose the optic

glass enters, as at H, and spreads it self all over the inside of the

eye, till within a small diftance from A and C. Now any object, as I K,

being placed before the eye, the rays of light issuing from each point

of this object are so refracted by the convex surface of the aqueous

humour, as to be caused to converge; after this being received by the

convex surface E D G of the crystalline humour, which has a greater

refractive power than the aqueous, the rays, when they are entered into

this surface, still more converge, and at going out of the surface E F

G into a humour of a less refractive power than the crystalline they

are made to converge yet farther. By all these successive refractions

they are brought to converge at the bottom of the eye, so that a

distinct image of the object as L M is impress’d on the nerve. And by

this means the object is seen.

11. IT has been made a difficulty, that the image of the object

impressed on the nerve is inverted, so that the upper part of the image

is impressed on the lower part of the eye. But this difficulty, I

think, can no longer remain, if we only consider, that upper and lower

are terms merely relative to the ordinary position of our bodies: and

our bodies, when view’d by the eye, have their image as much inverted

as other objects; so that the image of our own bodies, and of other

objects, are impressed on the eye in the same relation to one another,

as they really have.

12. THE eye can see objects equally distinct at very different

distances, but in one distance only at the same time. That the eye may

accomodate itself to different distances, some change in its humours

is requir’d. It is my opinion, that this change is made in the figure

of the crystalline humour, as I have indeavoured to prove in another

place.

13. IF any of the humours of the eye are too flat, they will refract

the light too little; which is the case in old age. If they are too

convex, they refract too much; as in those who are short-sighted.

14. THE manner of direct vision being thus explained, I proceed to give

some account of telescopes, by which we view more distinctly remote

objects; and also of microscopes, whereby we magnify the appearance of

small objects. In the first place, the most simple sort of telescope

is composed of two glasses, either both convex, or one convex, and the

other concave. (The first sort of these is represented in fig. 155, the

latter in fig. 156.)

15. IN fig. 155 let A B represent the convex glass next the object, C

D the other glass more convex near the eye. Suppose the object-glass A

B to form the image of the object at E F; so that if a sheet of white

paper were to be held in this place, the object would appear. Now

suppose the rays, which pass the glass A B, and are united about F, to

proceed to the eye glass C D, and be there refracted. Three only of

these rays are drawn in the figure, those which pass by the extremities

of the glass A B, and that which passes its middle. If the glass C D

be placed at such a distance from the image E F, that the rays, which

pass by the point F, after having proceeded through the glass diverge

so much, as the rays do that come from an object, which is at such a

distance from the eye as to be seen distinctly, these being received

by the eye will make on the bottom of the eye a distinct representation

of the point F. In like manner the rays, which pass through the object

glass A B to the point E after proceeding through the eye-glass C D

will on the bottom of the eye make a distinct representation of the

point E. But if the eye be placed where these rays, which proceed from

E, cross those, which proceed from F, the eye will receive the distinct

impression of both these points at the same time; and consequently

will also receive a distinct impression from all the intermediate

parts of the image E F, that is, the eye will see the object, to which

the telescope is directed, distinctly. The place of the eye is about

the point G, where the rays H E, H F cross, which pass through the

middle of the object-glass A B to the points E and F; or at the place

where the focus would be formed by rays coming from the point H, and

refracted by the glass C D. To judge how much this instrument magnifies

any object, we must first observe, that the angle under E H F, in which

the eye at the point H would see the image E F, is nearly the same as

the angle, under which the object appears by direct vision; but when

the eye is in G, and views the object through the telescope, it sees

the same under a greater angle; for the rays, which coming from E and F

cross in G, make a greater angle than the rays, which proceed from the

point H to these points E and F. The angle at G is greater than that at

H in the proportion, as the distance between the glasses A B and C D is

greater than the distance of the point G from the glass C D.

16. THIS telescope inverts the object; for the rays, which came from

the right-hand side of the object, go to the point E the left side of

the image; and the rays, which come from the left side of the object,

go to F the right side of the image. These rays cross again in G, so

that the rays, which come from the right side of the object, go to the

right side of the eye; and the rays from the left side of the object go

to the left side of the eye. Therefore in this telescope the image in

the eye has the same situation as the object; and seeing that in direct

vision the image in the eye has an inverted situation, here, where

the situation is not inverted, the object must appear so. This is no

inconvenience to astronomers in celestial observations; but for objects

here on the earth it is usual to add two other convex glasses, which

may turn the object again (as is represented in fig. 157,) or else to

use the other kind of telescope with a concave eye-glass.

17. IN this other kind of telescope the effect is founded on the same

principles, as in the former. The distinctness of the appearance is

procured in the same manner. But here the eye-glass C D (in fig. 156)

is placed between the image E F, and the object glass A B. By this

means the rays, which come from the right-hand side of the object,

and proceed toward E the left side of the image, being intercepted by

the eye-glass are carried to the left side of the eye; and the rays,

which come from the left side of the object, go to the right side of

the eye; so that the impression in the eye being inverted the object

appears in the same situation, as when view’d by the naked eye. The

eye must here be placed close to the glass. The degree of magnifying in

this instrument is thus to be found. Let the rays, which pass through

the glass A B at H, after the refraction of the eye-glass C D diverge,

as if they came from the point G; then the rays, which come from the

extremities of the object, enter the eye under the angle at G; so

that here also the object will be magnified in the proportion of the

distance between the glasses, to the distance of G from the eye-glass.

18. THE space, that can be taken in at one view in this telescope,

depends on the breadth of the pupil of the eye; for as the rays, which

go to the points E, F of the image, are something distant from each

other, when they come out of the glass C D, if they are wider asunder

than the pupil, it is evident, that they cannot both enter the eye

at once. In the other telescope the eye is placed in the point G,

where the rays that come from the points E or F cross each other, and

therefore must enter the eye together. On this account the telescope

with convex glasses takes in a larger view, than those with concave.

But in these also the extent of the view is limited, because the

eye-glass does not by the refraction towards its edges form so distinct

a representation of the object, as near the middle.

18. MICROSCOPES are of two sorts. One kind is only a very convex glass,

by the means of which the object may be brought very near the eye, and

yet be seen distinctly. This microscope magnifies in proportion, as

the object by being brought near the eye will form a broader impression

on the optic nerve. The other kind made with convex glasses produces

its effects in the same manner as the telescope. Let the object A B

(in fig. 158) be placed under the glass C D, and by this glass let an

image be formed of this object. Above this image let the glass G H be

placed. By this glass let the rays, which proceed from the points A and

B, be refracted, as is expressed in the figure. In particular, let the

rays, which from each of these points pass through the middle of the

glass C D, cross in I, and there let the eye be placed. Here the object

will appear larger, when seen through the microscope, than if that

instrument were removed, in proportion as the angle, in which these

rays cross in I, is greater than the angle, which the lines would make,

that should be drawn from I to A and B; that is, in the proportion made

up of the proportion of the distance of the object A B from I, to the

distance of I from the glass G H; and of the proportion of the distance

between the glasses, to the distance of the object A B from the glass C

D.

19. I SHALL now proceed to explain the imperfection in these

instruments, occasioned by the different refrangibility of the light

which comes from every object. This prevents the image of the object

from being formed in the focus of the object glass with perfect

distinctness; so that if the eye-glass magnify the image overmuch,

the imperfections of it must be visible, and make the whole appear

confused. Our author more fully to satisfy himself, that the different

refrangibility of the several sorts of rays is sufficient to produce

this irregularity, underwent the labour of a very nice and difficult

experiment, whose process he has at large set down, to prove, that the

rays of light are refracted as differently in the small refraction

of telescope glasses as in the larger of the prism; so exceeding

careful has he been in searching out the true cause of this effect.

And he used, I suppose, the greater caution, because another reason

had before been generally assigned for it. It was the opinion of all

mathematicians, that this defect in telescopes arose from the figure,

in which the glasses were formed; a spherical refracting surface not

collecting into an exact point all the rays which come from any one

point of an object, as has before been said[330]. But after our author

has proved, that in these small refractions, as well as in greater,

the sine of incidence into air out of glass, to the sine of refraction

in the red-making rays, is as 50 to 77, and in the blue-making rays 50

to 78; he proceeds to compare the inequalities of refraction arising

from this different refrangibility of the rays, with the inequalities,

which would follow from the figure of the glass, were light uniformly

refracted. For this purpose he observes, that if rays issuing from

a point so remote from the object glass of a telescope, as to be

esteemed parallel, which is the case of the rays, which come from the

heavenly bodies; then the distance from the glass of the point, in

which the least refrangible rays are united, will be to the distance,

at which the most refrangible rays unite, as 28 to 27; and therefore

that the least space, into which all the rays can be collected, will

not be less than the 55th part of the breadth of the glass. For if A

B (in fig. 159) be the glass, C D its axis, E A, F B two rays of the

light parallel to that axis entring the glass near its edges; after

refraction let the least refrangible part of these rays meet in G,

the most refrangible in H; then, as has been said, G I will be to I

H, as 28 to 27; that is, G H will be the 28th part of G I, and the

27th part of H I; whence if K L be drawn through G, and M N through H,

perpendicular to C D, M N will be the a 28th part of A B, the breadth

of the glass, and K L the 27th part of the same; so that O P the least

space, into which the rays are gathered, will be about half the mean

between these two, that is the 55th part of A B.

20. THIS is the error arising from the different refrangibility

of the rays of light, which our author finds vastly to exceed the

other, consequent upon the figure of the glass. In particular, if the

telescope glass be flat on one side, and convex on the other; when the

flat side is turned towards the object, by a theorem, which he has laid

down, the error from the figure comes out above 5000 times less than

the other. This other inequality is so great, that telescopes could

not perform so well as they do, were it not that the light does not

equally fill all the space O P, over which it is scattered, but is much

more dense toward the middle of that space than at the extremities. And

besides, all the kinds of rays affect not the sense equally strong, the

yellow and orange being the strongest, the red and green next to them,

the blue indigo and violet being much darker and fainter colours; and

it is shewn that all the yellow and orange, and three fifths of the

brighter half of the red next the orange, and as great a share of the

brighter half of the green next the yellow, will be collected into a

space whose breadth is not above the 250th part of the breadth of the

glass.

[Illustration]

And the remaining colours, which fall without this space, as they are

much more dull and obscure than these, so will they be likewise much

more diffused; and therefore call hardly affect the sense in comparison

of the other. And agreeable to this is the observation of astronomers,

that telescopes between twenty and sixty feet in length represent the

fixed stars, as being about 5 or 6, at most about 8 or 10 seconds in

diameter. Whereas other arguments shew us, that they do not really

appear to us of any sensible magnitude any otherwise than as their

light is dilated by refraction. One proof that the fixed stars do not

appear to us under any sensible angle is, that when the moon passes

over any of them, their light does not, like the planets on the same

occasion, disappear by degrees, but vanishes at once.

21. OUR author being thus convinced, that telescopes were not

capable of being brought to much greater perfection than at present

by refractions, contrived one by reflection, in which there is no

separation made of the different coloured light; for in every kind of

light the rays after reflection have the same degree of inclination

to the surface, from whence they are reflected, as they have at their

incidence, so that those rays which come to the surface in one line,

will go off also in one line without any parting from one another.

Accordingly in the attempt he succeeded so well, that a short one, not

much exceeding six inches in length, equalled an ordinary telescope

whose length was four feet. Instruments of this kind to greater

lengths, have of late been made, which fully answer expectation[331].

~CHAP. V.~

Of the RAINBOW.

I SHALL now explain the rainbow. The manner of its production was

understood, in the general, before Sir ~ISAAC NEWTON~ had discovered

his theory of colours; but what caused the diversity of colours in it

could not then be known, which obliges him to explain this appearance

particularly; whom we shall imitate as follows. The first person, who

expressly shewed the rainbow to be formed by the reflection of the

sun-beams from drops of falling rain, was ANTONIO DE DOMINIS. But this

was afterwards more fully and distinctly explained by DESCARTES.

2. THERE appears most frequently two rainbows; both of which are caused

by the foresaid reflection of the sun-beams from the drops of falling

rain, but are not produced by all the light which falls upon and are

reflected from the drops. The inner bow is produced by those rays only

which enter the drop, and at their entrance are so refracted as to

unite into a point, as it were, upon the farther surface of the drop,

as is represented in fig. 160; where the contiguous rays _a b_, _c d_,

_e f_, coming from the sun, and therefore to sense parallel, upon

their entrance into the drop in the points _b, d, f_, are so refracted

as to meet together in the point _g_, upon the farther surface of

the drop. Now these rays being reflected nearly from the same point

of the surface, the angle of incidence of each ray upon the point g

being equal to the angle of reflection, the rays will return in the

lines _g h, g k, g l_, in the same manner inclined to each other, as

they were before their incidence upon the point _g_, and will make

the same angles with the surface of the drop at the points _b, k,

l_, as at the points _b, d, f_, after their entrance; and therefore

after their emergence out of the drop each ray will be inclined to the

surface in the same angle, as when it first entered it; whence the

lines _b m, k n, l o_, in which the rays emerge, must be parallel to

each other, as well as the lines _a b, c d, e f_, in which they were

incident. But these emerging rays being parallel will not spread nor

diverge from each other in their passage from the drop, and therefore

will enter the eye conveniently situated in sufficient plenty to cause

a sensation. Whereas all the other rays, whether those nearer the

center of the drop, as _p q, r s_, or those farther off, as _t u, w

x_, will be reflected from other points in the hinder surface of the

drop; namely, the ray _p q_ from the point _y, r s_ from _z, t v_

from α, and _w x_ from β. And for this reason by their reflection and

succeeding refraction they will be scattered after their emergence from

the forementioned rays and from each other, and therefore cannot enter

the eye placed to receive them copious enough to excite any distinct

sensation.

3. THE external rainbow is formed by two reflections made between the

incidence and emergence of the rays; for it is to be noted, that the

rays _g h, g k, g l_, at the points _h, k, l_, do not wholly pass

out of the drop, but are in part reflected back; though the second

reflection of these particular rays does not form the outer bow. For

this bow is made by those rays, which after their entrance into the

drop are by the refraction of it united, before they arrive at the

farther surface, at such a distance from it, that when they fall

upon that surface, they may be reflected in parallel lines, as is

represented in fig. 161; where the rays _a b, c d, e f_, are collected

by the refraction of the drop into the point _g_, and passing on from

thence strike upon the surface of the drop in the points _h, k, l_, and

are thence reflected to _m, n, o_, passing from _h_ to _m_, from _k_

to _n_, and from _l_ to _o_ in parallel lines. For these rays after

reflection at _m, n, o_, will meet again in the point _p_, at the same

distance from these points of reflection _m, n, o_, as the point _g_ is

from the former points of reflection _h, k, l_. Therefore these rays

in passing from _p_ to the surface of the drop will fall upon that

surface in the points _q, r, s_ in the same angles, as these rays made

with the surface in _b, d, f_, after refraction. Consequently, when

these rays emerge out of the drop into the air, each ray will make

with the surface of the drop the same angle, as it made at its first

incidence; so that the lines _q t, r v, s w_, in which they come from

the drop, will be parallel to each other, as well as the lines _a b,

c d, e f_, in which they came to the drop. By this means these rays

to a spectator commodiously situated will become visible. But all the

other rays, as well those nearer the center of the drop _x y_, _z_

α, as those more remote from it β γ, δ ε, will be reflected in lines

not parallel to the lines _h m, k n, l o_; namely, the ray _x y_, in

the line ζ η, the ray ϰ α in the line θ ϰ, the ray β γ in the line

λ μ, and the ray δ ε in the line ν χ. Whence these rays after their

next reflection and subsequent refraction will be scattered from the

forementioned rays, and from one another, and by that means become

invisible.

4. IT is farther to be remarked, that if in the first case the incident

rays _a b, c d, e f_, and their correspondent emergent rays _h m, k

n, l o_, are produced till they meet, they will make with each other

a greater angle, than any other incident ray will make with its

corresponding emergent ray. And in the latter case, on the contrary,

the emergent rays _q t, r v, s w_ make with the incident rays an acuter

angle, than is made by any other of the emergent rays.

5. OUR author delivers a method of finding each of these extream angles

from the degree of refraction being given; by which method it appears,

that the first of these angles is the less, and the latter the greater,

by how much the refractive power of the drop, or the refrangibility of

the rays is greater. And this last consideration fully compleats the

doctrine of the rainbow, and shews, why the colours of each bow are

ranged in the order wherein they are seen.

6. SUPPOSE A (in fig. 162.) to be the eye, B, C, D, E, F, drops of

rain, M _n_, O _p_, Q _r_, S _t_, V _w_ parcels of rays of the sun,

which entring the drops B, C, D, E, F after one reflection pass out to

the eye in A. Now let M _n_ be produced to η till it meets with the

emergent ray likewise produced, let O _p_ produced meet its emergent

ray produced in ϰ, let Q _r_ meet its emergent ray in λ, let S _t_ meet

its emergent ray in μ, and let V _w_ meet its emergent ray produced

in ν. If the angle under M η A be that, which is derived from the

refraction of the violet-making rays by the method we have here spoken

of, it follows that the violet light will only enter the eye from

the drop B, all the other coloured rays passing below it, that is,

all those rays which are not scattered, but go out parallel so as to

cause a sensation. For the angle, which these parallel emergent rays

makes with the incident in the most refrangible or violet-making rays,

being less than this angle in any other sort of rays, none of the rays

which emerge parallel, except the violet-making, will enter the eye

under the angle M η A, but the rest making with the incident ray M η

a greater angle than this will pass below the eye. In like manner if

the angle under O ϰ A agrees to the blue-making rays, the blue rays

only shall enter the eye from the drop C, and all the other coloured

rays will pass by the eye, the violet-coloured rays passing above, the

other colours below. Farther, the angle Q λ A corresponding to the

green-making rays, those only shall enter the eye from the drop D, the

violet and blue-making rays passing above, and the other colours, that

is the yellow and red, below. And if the angle S μ A answers to the

refraction of the yellow-making rays, they only shall come to the eye

from the drop E. And in the last place, if the angle V ν A belongs to

the red-making and least refrangible rays, they only shall enter the

eye from the drop F, all the other coloured rays passing above.

7. BUT now it is evident, that all the drops of water found in any of

the lines A ϰ, A λ, A μ, A ν, whether farther from the eye, or nearer

than the drops B, C, D, E, F, will give the same colours as these do,

all the drops upon each line giving the same colour; so that the light

reflected from a number of these drops will become copious enough to be

visible; whereas the reflection from one minute drop alone could not be

perceived. But besides, it is farther manifest, that if the line A Ξ be

drawn from the sun through the eye, that is, parallel to the lines M

_n_, O _p_, Q _r_, S _t_, V _w_, and if drops of water are placed all

round this line, the same colour will be exhibited by all the drops at

the same distance from this line. Hence it follows, that when the sun

is moderately elevated above the horizon, if it rains opposite to it,

and the sun shines upon the drops as they fall, a spectator with his

back turned to the sun must observe a coloured circular arch reaching

to the horizon, being red without, next to that yellow, then green,

blue, and on the inner edge violet; only this last colour appears faint

by being diluted with the white light of the clouds, and from another

cause to be mentioned hereafter[332].

8. THUS is caused the interior or primary bow. The drops of rain at

some distance without this bow will cause the exterior or secondary

bow by two reflections of the sun’s light. Let these drops be G, H, I,

K, L; X _y_, Z α, Γ β, Δ ι, Θ ζ denoting parcels of rays which enter

each drop. Now it has been remarked, that these rays make with the

visible refracted rays the greatest angle in those rays, which are

most refrangible. Suppose therefore the visible refracted rays, which

pass out from each drop after two reflections, and enter the eye in

A, to intersect the incident rays in π, ρ, σ, τ, φ respectively. It

is manifest, that the angle under Θ φ A is the greatest of all, next

to that the angle under Δ τ A, the next in bigness will be the angle

under Γ σ A, the next to this the angle under Z ρ A, and the least of

all the angle under X π A. From the drop L therefore will come to the

eye the violet-making, or most refrangible rays, from K the blue, from

I the green, from H the yellow, and from G the red-making rays; and

the like will happen to all the drops in the lines A π, A ρ, A τ, A φ,

and also to all the drops at the same distances from the line A Ξ all

round that line. Whence appears the reason of the secondary bow, which

is seen without the other, having its colours in a contrary order,

violet without and red within; though the colours are fainter than in

the other bow, as being made by two reflections, and two refractions;

whereas the other bow is made by two refractions, and one reflection

only.

9. THERE is a farther appearance in the rainbow particularly described

about five years ago[333], which is, that under the upper part or

the inner bow there appears often two or three orders of very faint

colours, making alternate arches of green, and a reddish purple. At the

time this appearance was taken notice of, I gave my thoughts concerning

the cause of it[334], which I shall here repeat. Sir ~ISAAC NEWTON~ has

observed, that in glass, which is polished and quick-silvered, there is

an irregular refraction made, whereby some small quantity of light is

scattered from the principal reflected beam[335]. If we allow the same

thing to happen in the reflection whereby the rainbow is caused, it

seems sufficient to produce the appearance now mentioned.

10. LET A B (in fig. 162.) represent a globule of water, B the point

from whence the rays of any determinate species being reflected to C,

and afterwards emerging in the line C D, would proceed to the eye, and

cause the appearance of that colour in the rainbow, which appertains to

this species. Here suppose, that besides what is reflected regularly,

some small part of the light is irregularly scattered every way; so

that from the point B, besides the rays that are regularly reflected

from B to C, some scattered rays will return in other lines, as

in B E, B F, B G, B H, on each side the line B C. Now it has been

observed above[336], that the rays of light in their passage from one

superficies of a refracting body to the other undergo alternate fits

of easy transmission and reflection, succeeding each other at equal

intervals; insomuch that if they reach the farther superficies in one

sort of those fits, they shall be transmitted; if in the other kind

of them, they shall rather be reflected back. Whence the rays that

proceed from B to C, and emerge in the line C D, being in a fit of

easy transmission, the scattered rays, that fall at a small distance

without these on either side (suppose the rays that pass in the lines

B E, B G) shall fall on the surface in a fit of easy reflection, and

shall not emerge; but the scattered rays, that pass at some distance

without these last, shall arrive at the surface of the globule in a fit

of easy transmission, and break through that surface. Suppose these

rays to pass in the lines B F, B H; the former of which rays shall have

had one fit more of easy transmission, and the latter one fit less,

than the rays that pass from B to C. Now both these rays, when they

go out of the globule, will proceed by the refraction of the water

In the lines F I, H K, that will be inclined almost equally to the

rays incident on the globule, which come from the sun; but the angles

of their inclination will be less than the angle, in which the rays

emerging in the line C D are inclined to those incident rays. And after

the same manner rays scattered from the point B at a certain distance

without these will emerge out of the globule, while the intermediate

rays are intercepted; and these emergent rays will be inclined to the

rays incident on the globule in angles still less than the angles, in

which the rays F I and H K are inclined to them; and without these rays

will emerge other rays, that shall be inclined to the incident rays in

angles yet less.

[Illustration]

Now by this means may be formed of every kind of rays, besides the

principal arch, which goes to the formation of the rainbow, other

arches within every one of the principal of the same colour, though

much more faint; and this for divers successions, as long as these weak

lights, which in every arch grow more and more obscure, shall continue

visible. Now as the arches produced by each colour will be variously

mixed together, the diversity of colours observ’d in these secondary

arches may very possibly arise from them.

11. IN the darker colours these arches may reach below the bow, and

be seen distinct. In the brighter colours these arches are lost in

the inferior part of the principal light of the rainbow; but in all

probability they contribute to the red tincture, which the purple of

the rainbow usually has, and is most remarkable when these secondary

colours appear strongest. However these secondary arches in the

brightest colours may possibly extend with a very faint light below the

bow, and tinge the purple of these secondary arches with a reddish hue.

12. THE precise distances between the principal arch and these fainter

arches depend on the magnitude of the drops, wherein they are formed.

To make them any degree separate it is necessary the drop be exceeding

small. It is most likely, that they are formed in the vapour of the

cloud, which the air being put in motion by the fall of the rain may

carry down along with the larger drops; and this may be the reason, why

these colours appear under the upper part of the bow only, this vapour

not descending very low. As a farther confirmation of this, these

colours are seen strongest, when the rain falls from very black clouds,

which cause the fiercest rains, by the fall whereof the air will be

most agitated.

13. TO the like alternate return of the fits of easy transmission and

reflection in the passage of light through the globules of water, which

compose the clouds, Sir ISAAC NEWTON ascribes some of those coloured

circles, which at times appear about the sun and moon[337].

[Illustration]

[Illustration]

CONCLUSION.

SIR ~ISAAC NEWTON~ having concluded each of his philosophical treatises

with some general reflections, I shall now take leave of my readers

with a short account of what he has there delivered. At the end of

his mathematical principles of natural philosophy he has given us his

thoughts concerning the Deity. Wherein he first observes, that the

similitude found in all parts of the universe makes it undoubted, that

the whole is governed by one supreme being, to whom the original is

owing of the frame of nature, which evidently is the effect of choice

and design. He then proceeds briefly to state the best metaphysical

notions concerning God. In short, we cannot conceive either of space

or time otherwise than as necessarily existing; this Being therefore,

on whom all others depend, must certainly exist by the same necessity

of nature. Consequently wherever space and time is found, there God

must also be. And as it appears impossible to us, that space should be

limited, or that time should have had a beginning, the Deity must be

both immense and eternal.

2. AT the end of his treatise of optics he has proposed some thoughts

concerning other parts of nature, which he had not distinctly searched

into. He begins with some farther reflections concerning light, which

he had not fully examined. In particular he declares his sentiments at

large concerning the power, whereby bodies and light act on each other.

In some parts of his book he had given short hints at his opinion

concerning this[338], but here he expressly declares his conjecture,

which we have already mentioned[339], that this power is lodged in

a very subtle spirit of a great elastic force diffused thro’ the

universe, producing not only this, but many other natural operations.

He thinks it not impossible, that the power of gravity itself should be

owing to it. On this occasion he enumerates many natural appearances,

the chief of which are produced by chymical experiments. From numerous

observations of this kind he makes no doubt, that the smallest parts of

matter, when near contact, act strongly on each other, sometimes being

mutually attracted, at other times repelled.

3. THE attractive power is more manifest than the other, for

the parts of all bodies adhere by this principle. And the name of

attraction, which our author has given to it, has been very freely

made use of by many writers, and as much objected to by others. He has

often complained to me of having been misunderstood in this matter.

What he lays upon this head was not intended by him as a philosophical

explanation of any appearances, but only to point out a power in nature

not hitherto distinctly observed, the cause of which, and the manner of

its acting, he thought was worthy of a diligent enquiry. To acquiesce

in the explanation of any appearance by asserting it to be a general

power of attraction, is not to improve our knowledge in philosophy, but

rather to put a stop to our farther search.

FINIS.

[Illustration]

FOOTNOTES:

[1] Philosoph. Nat. princ. math. L. iii. introduct.

[2] Nov. Org. Scient. L. i. Aphorism. 9.

[3] Nov. Org. L. i. Aph. 19.

[4] Ibid. Aph. 25.

[5] Aph. 30. Errores radicales & in prima digestione mentis ab

excellentia functionum & remediorum sequentium non curantur.

[6] Aph. 38.

[7] Ibid.

[8] Aph. 39.

[9] Aph. 41.

[10] Aph. 10, 24.

[11] Aph. 45.

[12] De Cartes Princ. Phil. Part. 3. §. 52.

[13] Fermat, in Oper. pag. 156, &c.

[14] Nov. Org. Aph. 46.

[15] Aph. 50.

[16] Ibid.

[17] Aph 53.

[18] Aph. 54.

[19] Aph. 56.

[20] Aph. 55.

[21] Locke, On human understanding, B. iii.

[22] Nov. Org. Aph. 59.

[23] In the conclusion.

[24] Nov. Org. L. i. Aph. 59.

[25] Ibid. Aph. 60.

[26] Ibid. Aph. 62.

[27] Aph. 63.

[28] Aph. 64.

[29] Aph. 65.

[30] See above, § 4, 5.

[31] Nov. Org. L. i. Aph. 69.

[32] Ibid.

[33] Ibid. Aph. 109.

[34] Book III. Chap. iv.

[35] Book I. Chap. 2. § 14.

[36] Ibid. § 85, &c.

[37] See Book II. Ch. 3. § 3, 4. of this treatise.

[38] See Book II. Ch. 3. of this treatise.

[39] See Chap. 4.

[40] At the end of his Optics. in Qu. 21.

[41] See the same treatise, in Advertisement 2.

[42] Nov. Org. Lib. i. Ax. 105.

[43] Princip. philos. pag. 13, 14.

[44] Princ. Philos. L. II. prop. 24. corol. 7. See also B. II. Ch. 5. §

3. of this treatise.

[45] How this degree of elasticity is to be found by experiment, will

be shewn below in § 74.

[46] In oper. posthum de Motu corpor. ex percussion. prop. 9.

[47] In the above-cited place.

[48] In the place above-cited.

[49] These experiments are described in § 73.

[50] Book II. Chap. 5.

[51] Chap. 1. § 25, 26, 27, compared with § 15, &c.

[52] Book II. Chap. 5. § 3.

[53] See Euclid’s Elements, Book XII. prop. 13.

[54] Archimed. de æquipond. prop. 11.

[55] Ibid. prop. 12.

[56] Lucas Valerius De centr. gravit. solid. L. I. prop. 2.

[57] Idem L. II. prop. 2.

[58] § 25.

[59] § 27.

[60] Pag. 65, 68.

[61] § 23.

[62] § 20

[63] § 17.

[64] § 27.

[65] Hugen. Horolog. oscillat. pag. 141, 142.

[66] See Hugen. Horolog. Oscillat. p. 142.

[67] Princip. Philos. pag. 22.

[68] Chap. 1. § 29.

[69] Princip. Philos. pag. 25.

[70] § 71.

[71] See Method. Increment. prop. 25.

[72] Lib. XI. Def.

[73] Chap. 2. § 17.

[74] See above Ch. 2. § 17.

[75] From B II. Ch. 3.

[76] Prin. Philos. pag. 7, &c.

[77] See Newton, princip. philos. pag. 9. lin. 30.

[78] Princip. Philos. pag. 10.

[79] Renat. Des Cart. Princ. Philos. Part. II. § 25.

[80] Ibid. § 30.

[81] § 85, &c.

[82] Princip. Philos. Lib. I. prop. 9.

[83] § 92.

[84] Ch. II. § 22.

[85] Viz. L. I. prop. 30, 29, & 26.

[86] Ch. II. § 21, 22.

[87] viz. His doctrine of prime and ultimate ratios.

[88] § 57

[89] § 3.

[90] Ch. 2. § 22.

[91] § 12.

[92] Ch. 1. sect. 21, 22.

[93] Elem. Book I. p. 37.

[94] § 12.

[95] Ch 1 § 24.

[96] Ch 2 select. 17.

[97] Newt. Princ. L. II. prop. 2; 5, 6, 7; 11, 12.

[98] Prop. 3; 8, 9; 13, 14.

[99] Prop. 4.

[100] Prælect. Geometr. pag. 123.

[101] Newton. Princ. Lib. II. prop. 10.

[102] Newton. Princ. Lib II. prop 10. in schol.

[103] Torricelli de motu gravium.

[104] Ch. 2 § 85, &c.

[105] Newt. Princ L. II. sect 6.

[106] L. II. sect. 4.

[107] See B. II. Ch 6. § 7. of this treatise.

[108] Lib. I. sect. 10.

[109] De la Pesanteur, pag. 169, and the following.

[110] Newton. Princ. L. II. prop 4. schol.

[111] See his Tract on the admirable rarifaction of the air.

[112] Book II. Ch. 6.

[113] Princ. philos. Lib. II. prop. 23.

[114] Book I. Ch. 2. § 30.

[115] Princ. philos. Lib. II. prop. 23, in schol.

[116] Princ. philos. Lib. II. prop. 33. coroll.

[117] Lib. II. Ch. 5.

[118] Ibid. Prop. 35. coroll. 2.

[119] Ibid. coroll. 3.

[120] Vid. ibid. coroll. 6.

[121] In § 2.

[122] Princ. philos. Lib. II. Prop. 35.

[123] Ibid.

[124] Id.

[125] h. 1. § 29.

[126] Princ. philos. Lib. II. Prop. 38, compared with coroll. 1 of

prop. 35.

[127] L. II. Lem. 7. schol. pag. 341.

[128] Lib. II. Prop. 34.

[129] Lib. II. Lem. 7. p. 341.

[130] Schol. to Lem. 7.

[131] Prop. 34. schol.

[132] Ibid.

[133] Ibid.

[134] Book II. Ch. I. § 6.

[135] Vid. Newt. princ. in schol. to Lem. 7, of Lib. II. pag. 341.

[136] Sect. 17. of this chapter.

[137] See Princ. philos. Lib. II. prop. 34.

[138] Vid. Princ. philos. Lib. II. Lem. 5. p. 314.

[139] Lemm. 6.

[140] Ibid. 7.

[141] Newt. Princ. Lib. II. prop. 40, in schol.

[142] Lib. II. in schol. post prop. 31.

[143] Book I. ch. 2 § 82.

[144] Book I. Ch. 3 § 29.

[145] Ch. 3. of this present book.

[146] Ch. 4.

[147] In Princ. philos. part. 3.

[148] Philos. princ. mathem. Lib. II. prop. 2. & schol.

[149] Ibid. prop 53.

[150] Philos. princ. prop. 52. coroll. 4.

[151] Ibid.

[152] Coroll. 11.

[153] See ibid. schol. post prop. 53.

[154] Princ. philos. pag. 316, 317.

[155] Ch. I. § 7.

[156] Book I. Ch. 3.

[157] Book I. Ch. 3. § 29.

[158] Ibid. Ch. 2. § 30, 17.

[159] Book I. Ch. 3.

[160] Ch. 1. § 7.

[161] Chap. 5. § 8.

[162] Princ. pag. 60.

[163] Street, in Astron. Carolin.

[164] See Chap. 5. §9, &c.

[165] In the foregoing page.

[166] See Newton. Princ. Lib. III. prop. 13.

[167] Chap. 5. § 10.

[168] Princ. Lib. I. prop. 60.

[169] Book I, Chap. 2. § 80.

[170] Princ. philos. Lib. I. prop. 58. coroll. 3.

[171] Newt. Optics. pag. 378.

[172] Newton. Princ. Lib. III. prop. 1.

[173] Newton, Princ. Lib. III. pag. 390,391. compared with pag. 393.

[174] Book I. Ch. 3. § 29.

[175] Princ. philos. Lib. I. prop. 4.

[176] Ibid. coroll.

[177] Newt. Princ. philos. Lib. III. pag. 390.

[178] Newt. Princ. philos. Lib. III. pag. 391, 392.

[179] Book III. Ch. 4.

[180] Newt. Princ. philos. Lib. III. pag. 391.

[181] Ibid. pag. 392.

[182] See Book I. Ch. 2. § 60, 64.

[183] Book I. Ch. 2. § 17.

[184] See Ch. II. § 6.

[185] The second of the laws of motion laid down in Book I. Ch. 1.

[186] Newton. Princ. philos. Lib. III. prop. 6. pag. 401.

[187] Newton’s Princ. philos. Lib. III. prop. 22, 23.

[188] Newton. Princ. Lib. I. prop. 66. coroll. 7.

[189] Menelai Sphaeric. Lib. I. prop. 10.

[190] Vid. Newt. Princ. Lib. I. prop. 66. coroll. 10.

[191] Vid. Newt. Princ. Lib. III prop. 30. p. 440.

[192] Ibid. Lib. I. prop. 66. coroll. 10.

[193] What this proportion is, may be known from Coroll. 2 prop. 44.

Lib. I. Princ. philos. Newton.

[194] Princ. Phil. Newt. Lib. I. prop. 45. Coroll. 1.

[195] Pr. Phil. Newt. Lib. I. prop. 66. Coroll. 7.

[196] See § 19 of this chapter.

[197] Phil. Nat. Pr. Math Lib. I. prop. 66. cor. 8.

[198] Ibid. Coroll. 8.

[199] Ibid.

[200] Ibid.

[201] Newt. Princ. Lib. III. prop. 29.

[202] Ibid. prop. 28.

[203] Ibid. prop. 31.

[204] Newt. Princ. pag. 459.

[205] In Princ. philos. part. 3. § 41.

[206] Chap. 1. § 11.

[207] Newton. Princ. philos. Lib. III. Lemm. 4. pag. 478.

[208] Princ. philos. Lib. III. prop. 40.

[209] Book I. chap. 2. § 82.

[210] Princ. philos. Lib. III. pag. 499, 500.

[211] Ibid. pag. 500, and 520, &c.

[212] Princ. Philos. Lib. III. prop. 40.

[213] Ibid. prop. 41.

[214] Ibid. pag. 522.

[215] Ibid. prop. 42.

[216] Newt. Princ. philos. edit. 2. p. 464, 465.

[217] Ibid. edit. 3. p 501, 502.

[218] Ibid. pag. 519.

[219] Ibid. pag. 524.

[220] Newt. Princ. philos. p. 525.

[221] Ibid.

[222] Ibid. pag. 508.

[223] Ibid.

[224] Ibid. pag. 484.

[225] Ibid. pag. 482, 483.

[226] Ibid. pag. 481.

[227] Ibid. pag. 509.

[228] See the fore-cited place.

[229] Ibid. and Cartes. Princ. Phil. part. 3. § 134, &c.

[230] Vid. Phil. Nat. princ. Math. p. 511.

[231] Book I. Ch. 4. § 11.

[232] Ch. 5.

[233] All these arguments are laid down in Philos. Nat. Princ. Lib.

III. from p. 509, to 517.

[234] Philos. Nat. Princ. Lib. III. p. 515.

[235] Ch. 5.

[236] See Ch. 1. § 11.

[237] Newt. Princ. Philos. pag. 525, 526. An account of all the stars

of both these kinds, which have appeared within the last 150 years may

be seen in the Philosophical transactions, vol. 29. numb. 346.

[238] Newt. Princ. Philos. Nat. Lib. III. prop. 6.

[239] Ch. 3. § 6.

[240] Book I. Ch. 2. § 24.

[241] Newt. Princ. Lib. III. prop. 6.

[242] Ch. 3. § 6.

[243] Newt. Princ. philos. Lib. III. prop. 7. cor. 1.

[244] See Book I. Ch. 1. § 15.

[245] Ibid. § 5, 6.

[246] Chap. 2. § 8.

[247] Newt. Princ. Lib. I. prop. 63.

[248] § 8.

[249] See Introd. § 23.

[250] § 4, 5.

[251] Newt. Princ. philos. Lib. I. prop. 74.

[252] Ibid. coroll. 3.

[253] Lib. I. Prop. 75. and Lib. III. prop. 8.

[254] Lib. I. Prop. 76.

[255] Ibid. cor. 5.

[256] Vid. Lib. III. Prop. 7. coroll. 1

[257] Newt. Princ. Lib. III. prop. 8. coroll. 1.

[258] Ibid. coroll. 2.

[259] Book I. Ch. 4. § 2.

[260] Newt. Princ. Lib. III. prop. 8. coroll. 3.

[261] Ibid. coroll. 4.

[262] Book I. Ch. 4.

[263] Lib. II. prop. 20. cor. 2.

[264] Chap. 4. § 17.

[265] Ibid.

[266] Vid. Newt. Princ. Lib. II. prop. 46.

[267] Princ. philos. Lib. II. prop. 49.

[268] Chap. 3. § 18.

[269] Newt. Princ. philos. Lib. I. prop. 66. coroll. 18.

[270] § 8.

[271] Ch. 3. § 5.

[272] Ch. 3 § 17.

[273] Ibid.

[274] See below § 44.

[275] Newton Princ. Lib. III. prop. 19.

[276] Lib. III. prop. 19.

[277] Lib. I. prop. 73.

[278] Lib. III. prop. 20.

[279] Ibid.

[280] Opt. B. I. part. 2. prop. 1.

[281] Newt. Opt. B. 1. part 1. experim. 5.

[282] Ibid. prop. 4.

[283] Newt. Opt. B. 1. part 2. exper. 5.

[284] Ibid exper. 6.

[285] Newton Opt. B. I. prop. 10.

[286] Ibid exp. 9.

[287] Newt. Opt. B. I. part 1. exp 15.

[288] Philos. Transact. N. 88, p. 5099.

[289] Opt B. I. par. 2. exp. 14.

[290] Ibid. exp. 10.

[291] Opt. pag. 122.

[292] Opt. B. I. part 2. exp. 11.

[293] Ibid prop. 4, 6.

[294] Opt. pag. 51.

[295] Opt. Book II. prop. 8.

[296] Opt. Book II. par. 3. prop. 2.

[297] § 17.

[298] Opt. Book II. par. 3. prop. 4.

[299] Opt. Book II. pag. 241.

[300] Ibid. pag. 224.

[301] Ibid. Obs. 17. &c.

[302] Ibid. Obs. 10.

[303] Ibid. pag. 206.

[304] Obser. 21.

[305] Observ. 5. compared with Observ. 10

[306] Ibid. prop. 5.

[307] Observ. 7.

[308] Observ. 9.

[309] Ibid prop. 7.

[310] Opt. pag. 243.

[311] Newt. Opt. B. I. part. 1. prop. I.

[312] Opt. B. I. part. 1. prop. 2.

[313] Opt. B. I. part 1. Expec. 6.

[314] Opt. pag. 67, 68, &c.

[315] Ibid. B. 1. par. 2. prop. 3.

[316] Opt. B. II. par. 3. prop. 10.

[317] Opt. B. II. par. 3. prop. 15.

[318] Ibid. par. 1. observ. 7.

[319] Ibid. Observ. 19.

[320] Opt. B. II. par. 2. pag. 199. &c.

[321] Ibid. par. 4

[322] Ibid. part. 3. prop. 13.

[323] Ibid. prop. 17.

[324] Ibid. prop. 13.

[325] Opt. Qu. 18, &c.

[326] See Concl. S. 2.

[327] B. II. Ch. 1.

[328] Opt. B. III. Obs. 1.

[329] Ibid. Obs. 2.

[330] § 2.

[331] Philos. Trans. No. 378.

[332] § 11.

[333] Philos. Transact No. 375.

[334] Ibid.

[335] Opt. B. II. part 4.

[336] Ch. 3. § 14.

[337] Opt. B. II. part 4. obs. 13.

[338] Opt. pag. 255.

[339] Ch. 3. § 18.