the Moon, or the Semi-diameter of the Moon's Orbit, be 100000; then
shall its greatest Eccentricity TB be 66782 such Parts; and the least
TA, 43319. So that the greatest Equation of the Orbit, _viz._ when the
Apogee is in the Syzygys, will be 7 degrees, 39 minutes, 30 seconds, or
perhaps 7 degrees, 40 minutes, (for I suspect there will be some
Alteration, according to the Position of the Apogee in _Cancer_ and
_Capricorn_.) But when it is Quadrate to the Sun, the greatest Equation
aforesaid will be 4 degrees, 57 minutes, 56 seconds; and the greatest
Equation of the Apogee, 12 degrees, 15 minutes, 4 seconds.
Having from these Principles made a Table of the Equation of the Moon's
Apogee, and of the Eccentricities of her Orbit to each degree of the
Annual Argument, from whence the Eccentricity TF, and the Angle BTF
(_viz._ the second and the principal Equation of the Apogee) may easily
be had for any Time required; let the Equation thus found be added to
the first Equated Place of the Moon's Apogee, if the Annual Argument be
less than 90 degrees, or greater than 180 degrees, and less than 270;
otherwise it must be subducted from it; and the Sum or Difference shall
be the Place of the Lunar Apogee secondarily equated; which being taken
from the Moon's Place equated a third time, shall leave the mean Anomaly
of the Moon corresponding to any given Time. Moreover, from this mean
Anomaly of the Moon, and the before-found Eccentricity of her Orbit, may
be found (by means of a Table of Equations of the Moon's Centre made to
every degree of the mean Anomaly, and some Eccentricities, _viz._ 45000,
50000, 55000, 60000, and 65000) the _Prostaphæresis_, or Equation of the
Moon's Centre, as in the common way: And this being taken from the
former Semi-circle of the middle Anomaly, and added in the latter to the
Moon's Place thus thrice equated, will produce the Place of the Moon a
fourth time equated.
The greatest Variation of the Moon (_viz._ that which happens when the
Moon is in an Octant with the Sun) is nearly, reciprocally as the Cube
of the Distance of the Sun from the Earth. Let that be taken 37 minutes,
25 seconds, when the Sun is _in Perigæo_, and 33 minutes, 40 seconds,
when he is _in Apogæo_: And let the Differences of this Variation in the
Octants be made reciprocally, as the Cubes of the Distances of the Sun
from the Earth; and so let a Table be made of the aforesaid Variation of
the Moon in her Octants (or its Logarithms) to every Tenth, Sixth, or
Fifth Degree of the mean Anomaly: And for the Variation out of the
Octants, make, as Radius to the Sine of the double Distance of the Moon
from the next Syzygy, or Quadrature :: so let the afore-found Variation
in the Octant be to the Variation congruous to any other Aspect; and
this added to the Moon's Place before found in the first and third
Quadrant (accounting from the Sun) or subducted from it in the second
and fourth, will give the Moon's Place equated a fifth time.
Again, as Radius to the Sine of the Summ of the Distances of the Moon
from the Sun, and of her Apogee from the Sun's Apogee (or the Sine of
the Excess of that Summ above 360 degrees,) :: so is 2 minutes, 10
seconds, to a sixth Equation of the Moon's Place, which must be
subtracted, if the aforesaid Summ or Excess be less than a Semi-circle;
but added, if it be greater. Let it be made also, as Radius to the Sine
of the Moon's distance from the Sun :: so 2 degrees, 20 secants, to a
seventh Equation; which when the Moon's Light is increasing, add; but
when decreasing, subtract; and the Moon's Place will be equated a
seventh time, and this is her Place _in her proper Orbit_.
Note here, the Equation thus produced by the mean Quantity 2 degrees, 20
seconds, is not always of the same magnitude; but is increased and
diminished, according to the Position of the Lunar Apogee. For if the
Moon's Apogee be in Conjunction with the Sun's, the aforesaid Equation
is about 54 seconds greater: But when the Apogees are in Opposition,
'tis about as much less; and it librates between its greatest Quantity 3
minutes, 14 seconds, and its least, 1 minute, 26 seconds. And this is,
when the Lunar Apogee is in Conjunction, or Opposition with the Sun's:
But in the Quadratures, the aforesaid Equation is to be lessen'd about
50 seconds, or 1 minute, when the Apogees of the Sun and Moon are in
Conjunction; but if they are in Opposition, for want of a sufficient
number of Observations, I cannot determine, whether it is to be lessen'd
or increas'd. And even as to the Argument or Decrement of the Equation,
2 minutes, 20 seconds, above mentioned, I dare determine nothing
certain, for the same Reason, _viz._ the want of Observations accurately
made.
If the sixth and seventh Equations are augmented or diminished in a
reciprocal _Ratio_ of the distance of the Moon from the Earth; _i. e._
in a direct _Ratio_ of the Moon's Horizontal Parallax, they will become
more accurate: And this may be readily done, if Tables are first made to
each minute of the said Parallax, and to every sixth or fifth degree of
the Argument of the sixth Equation for the Sixth, as of the distance of
the Moon from the Sun, for the Seventh Equation.
From the Sun's Place, take the mean motion of the Moon's ascending Node,
equated as above; the Remainder shall be the Annual Argument of the
Node, whence its second Equation may be computed after the following
manner in the preceding Figure.
Let T, as before, represent the Earth; TS a Right Line, conjoining the
Earth and Sun: Let also the Line TACB, be drawn to the Place of the
ascending Node of the Moon, as above equated; and let STA be the Annual
Argument of the Node. Take TA from a Scale, and let it be to AB :: as 56
to 3, or as 11⅔ to 1. Then bissect BA in C, and on C as a Centre,
with the Distance CA, describe a Circle, as AFB, and make the Angle BCF,
equal to double the Annual Argument of the Node before-found: So shall
the Angle BTF, be the second Equation of the ascending Node; which must
be added, when the Node is passing from the Quadrature to a Syzygy with
the Sun; and subducted, when the Node moves from a Syzygy towards a
Quadrature. By which means, the true Place of the Node of the Lunar
Orbit will be gained: Whence from Tables made after the common way, the
_Moon's Latitude, and the Reduction of her Orbit to the Ecliptick_, may
be computed, supposing the Inclination of the Moon's Orbit to the
Ecliptick, to be 4 degrees, 59 minutes, 35 seconds, when the Nodes are
in Quadrature with the Sun; and 5 degrees, 17 minutes, 20 seconds, when
they are in the Syzygys.
And from the Longitude and Latitude thus found, and the given Obliquity
of the Ecliptick, 23 degrees, 29 minutes, the Right Ascension and
Declination of the Moon will be found.
The Horizontal Parallax of the Moon, when she is in the Syzygys, at a
mean distance from the Earth, I make to be 57 minutes, 30 seconds; and
her Horary Motion, 33 minutes, 32 seconds, 32 thirds; and her apparent
Diameter 31 minutes, 30 seconds. But in her Quadratures at a mean
Distance from the Earth, I make the Horizontal Parallax of the Moon to
be 59 minutes, 40 seconds, her Horary Motion 32 minutes, 12 seconds, 2
thirds, and her apparent Diameter, 31 minutes, 3 seconds. The Moon in an
Octant to the Sun, and at a mean distance, hath her Centre distant from
the Centre of the Earth about 60-2/9 of the Earth's Semi-diameters.
The Sun's Horizontal Parallax I make to be 10 seconds, and its apparent
Diameter at a mean distance from the Earth, I make 32 minutes, 15
seconds.
The Atmosphere of the Earth, by dispersing and refracting the Sun's
Light, casts a Shadow, as if it were an Opake Body, at least to the
height of 40 or 50 Geographical Miles (by a Geographical Mile, I mean
the sixtieth part of a Degree of a great Circle, on the Earth's
Surface.) This Shadow falling upon the Moon in a Lunar Eclipse, makes
the Earth's Shadow be the larger or broader. And to each Mile of the
Earth's Atmosphere, is correspondent a Second in the Moon's Disk, so
that the Semi-diameter of the Earth's shadow projected upon the Disk of
the Moon, is to be increased about 50 seconds: Or, which is all one, in
a Lunar Eclipse, the Horizontal Parallax of the Moon is to be increased
in the Ratio of about 70 to 69.
Thus far the Theory of this Incomparable Mathematician. And if we had
many Places of the Moon accurately observ'd, especially about her
Quadratures, and these well compar'd with her Places, at the same time
calculated according to this Theory; it would then appear, whether there
yet remain any other sensible Equations; which when accounted for, might
serve to improve and enlarge this Theory.
_An Estimate of the Degrees of the _Mortality_ of Mankind, drawn from
curious _Tables_ of the _Births_ and _Funerals_ at the City of
_Breslaw_; with an Attempt to ascertain the Price of _Annuities_ upon
_Lives_. By Mr. _E. Halley_, R. S. S._
The Contemplation of the _Mortality_ of _Mankind_, has besides the
_Moral_, its _Physical_ and _Political_ Uses, both which have been some
Years since most judiciously consider'd by the Curious Sir _William
Petty_, in his _Natural_ and _Political_ Observations on the Bills of
_Mortality_ of _London_, own'd by Captain _John Graunt_: And since in a
like Treatise on the Bills of _Mortality_ of _Dublin_. But the Deduction
from those Bills of _Mortality_ seemed even to their Authors to be
defective: First, In that the _Number_ of the People was wanting.
Secondly, That the _Ages_ of the People dying was not to be had. And
Lastly, That both _London_ and _Dublin_, by reason of the great and
casual Accession of _Strangers_ who die therein, (as appeared in both,
by the great Excess of the _Funerals_ above the _Births_) rendred them
incapable of being Standards for this purpose; which requires, if it
were possible, that the People we treat of, should not at all be
changed, but die where they were born, without any adventitious Increase
from Abroad, or Decay by Migration elsewhere.
[Illustration: _Plate 3. pag. 280._]
This _Defect_ seems in a great measure to be satisfied by the late
curious Tables of the Bills of _Mortality_ at the City of _Breslaw_,
lately communicated to this Honourable Society by Mr. _Justell_, wherein
both the Ages and Sexes of all that die, are Monthly delivered, and
compared with the number of the _Births_, for Five Years last past,
_viz._ 1687, 88, 89, 90, 91, seeming to be done with all the Exactness
and Sincerity possible.
This City of _Breslaw_ is the Capital City of the Province of _Silesia_;
or, as the _Germans_ call it, _Schlesia_, and is situated on the Western
Bank of the River _Oder_, anciently call'd _Viadrus_, near the Confines
of _Germany_ and _Poland_, and very nigh the Latitude of _London_. It is
very far from the Sea, and as much a _Mediterranean_ Place as can be
desired, whence the Confluence of Strangers is but small, and the
Manufacture of Linnen employs chiefly the poor People of the Place, as
well as of the Country round about; whence comes that sort of Linnen we
usually call your _Sclesiæ Linnen_; which is the chief, if not the only
Merchandize of the Place. For these Reasons, the People of this City
seem most proper for a _Standard_; and the rather, for that the _Births_
do a small matter exceed the Funerals. The only thing wanting, is the
Number of the whole People, which in some measure I have endeavour'd to
supply, by the comparison of the _Mortality_ of the People of all Ages,
which I shall from the said Bills trace out with all the Accuracy
possible.
It appears that in the Five Years mentioned, _viz._ from 87 to 91
inclusive, there were born 6193 Persons, and buried 5869; that is, born
_per Annum_ 1238, and buried 1174; whence an _Increase_ of the People
may be argued of 64 _per Annum_, or of about a 20th part, which may
perhaps be balanc'd by the Levies for the _Emperor_'s Service in his
Wars. But this being contingent, and the Births certain, I will suppose
the People of _Breslaw_ to be increased by 1238 _Births_ annually. Of
these it appears by the same Tables, that 348 do die _yearly_ in the
_first Year_ of their _Age_, and that but 890 do arrive at a full
_Year's Age_; and likewise, that 198 do die in the _Five Years_ between
1 and 6 compleat, taken at a _Medium_; so that but 692 of the Persons
_born_ do survive _Six_ whole _Years_. From this _Age_ the Infants being
arrived at some degree of Firmness, grow less and less _Mortal_; and it
appears, that of the whole People of _Breslaw_ there die _yearly_, as in
the following Table, wherein the upper Line shews the _Age_, and the
next under it, the _Number_ of Persons of that Age _dying yearly_.
7 8 9 14 18 21 27 28 35
11 11 6 5½ 2 3½ 5 6 4½ 6½ 9 8 7 7
36 42 45 39 54 55 56 63
8 9½ 8 9 7 7 10 11 9 9 10 12
70 71 72 77 81 84 90 91
9½ 14 9 11 9½ 6 7 3 4 2 1 1 1
98 99 100
0 ⅕ ⅗
And where no Figure is placed over, it is to be understood of those that
die between the Ages of the precedent and consequent _Column_.
From this Table it is evident, that from the Age of 9 to about 25, there
does not die above 6 _per Annum_ of each Age, which is much about 1 _per
Cent._ of those that are of those _Ages_: And whereas in the 14, 15, 16,
17 _Years_, there appear to die much fewer, as 2 and 3½; yet that
seems rather to be attributed to Chance, as are the other Irregularities
in the Series of Ages, which would rectifie themselves, were the number
of Years much more considerable, as 20 instead of 5. And by our own
Experience in _Christ-Church Hospital_, I am inform'd there die of the
_Young Lads_, much about 1 _per Cent. per Annum_, they being of the
aforesaid _Ages_. From 25 to 50, there seem to die from 7 to 8 and 9
_per Annum_ of each Age; and after that to 70, they growing more
_crasie_, though the number be much diminished, yet the _Mortality
increases_, and there are found to die 10 or 11 of each Age _per Annum_:
From thence the number of the Living being grown very small, they
gradually decline till there be none left to _die_; as may be seen at
one View in the Table.
From these Considerations I have form'd the _adjoined Table_, whose Uses
are manifold, and give a more just _Idea_ of the _State_ and _Condition_
of _Mankind_, than any thing yet extant that I know of. It exhibits the
_Number_ of _People_ in the City of _Breslaw_ of all Ages, from the
_Birth_ to extreme _Old Age_, and thereby shews the Chances of
_Mortality_ at all _Ages_, and likewise how to make a certain Estimate
of the Value of _Annuities_ for _Lives_, which hitherto has been only
done by an imaginary _Valuation_: Also the _Chances_ that there are that
a _Person_ of any _Age_ proposed does live to any other _Age_ given;
with many more, as I shall hereafter shew. This _Table_ does shew the
_Number_ of Persons that are living in the _Age_ current annexed
thereto, as follows:
+------+---------+------+---------+------+---------+
| Age. | Persons.| Age. | Persons.| Age. | Persons.|
| Curt.| | Curt.| | Curt.| |
+------+---------+------+---------+------+---------+
| 1 | 1000 | 8 | 680 | 15 | 628 |
| 2 | 855 | 9 | 670 | 16 | 622 |
| 3 | 798 | 10 | 661 | 17 | 616 |
| 4 | 760 | 11 | 653 | 18 | 610 |
| 5 | 732 | 12 | 646 | 19 | 604 |
| 6 | 710 | 13 | 640 | 20 | 598 |
| 7 | 692 | 14 | 634 | 21 | 592 |
+------+---------+------+---------+------+---------+
| 22 | 586 | 29 | 539 | 36 | 481 |
| 23 | 579 | 30 | 531 | 37 | 472 |
| 24 | 573 | 31 | 523 | 38 | 463 |
| 25 | 567 | 32 | 515 | 39 | 454 |
| 26 | 560 | 33 | 507 | 40 | 445 |
| 27 | 553 | 34 | 499 | 41 | 436 |
| 28 | 546 | 35 | 490 | 42 | 427 |
+------+---------+------+---------+------+---------+
| 43 | 417 | 50 | 346 | 57 | 272 |
| 44 | 407 | 51 | 335 | 58 | 262 |
| 45 | 397 | 52 | 324 | 59 | 252 |
| 46 | 387 | 53 | 313 | 60 | 242 |
| 47 | 377 | 54 | 302 | 61 | 232 |
| 48 | 367 | 55 | 292 | 62 | 222 |
| 49 | 357 | 56 | 282 | 63 | 212 |
+------+---------+------+---------+------+---------+
| 64 | 202 | 71 | 131 | 78 | 58 |
| 65 | 192 | 72 | 120 | 79 | 49 |
| 66 | 182 | 73 | 109 | 80 | 41 |
| 67 | 172 | 74 | 98 | 81 | 34 |
| 68 | 162 | 75 | 88 | 82 | 28 |
| 69 | 152 | 76 | 78 | 83 | 23 |
| 79 | 142 | 77 | 68 | 84 | 20 |
+------+---------+------+---------+------+---------+
Age. Persons.
7 5547
14 4584
21 4270
28 3964
35 3604
42 3708
49 2709
56 2194
63 1694
70 1204
77 692
84 253
100 107
-----------
34000
-----------
Sum Total.
Thus it appears, that the whole People of _Breslaw_ does consist of
34000 _Souls_, being the Sum _Total_ of the Persons of all Ages in the
_Table_: The first use hereof is to shew the Proportion of _Men_ able to
bear _Arms_ in any _Multitude_, which are those between 18 and 56,
rather than 16 and 60; the one being generally too weak to bear the
_Fatigues_ of _War_, and the Weight of _Arms_; and the other too crasie
and infirm from _Age_, notwithstanding particular Instances to the
contrary. Under 18 from the _Table_, are found in this City 11997
Persons, 3950 above 56, which together make 15947, so that the Residue
to 34000 being 18053, are Persons between those _Ages_. At least one
half thereof are Males, or 9027: So that the whole Force this City can
raise of _Fencible Men_, as the _Scotch_ call them, is about 9000, or
9/34, or somewhat more than a quarter of the _Number_ of _Souls_; which
may parhaps pass for a Rule for all other places.
The _Second Use_ of this _Table_, is, to shew the differing degrees of
_Mortality_, or rather _Vitality_, in all _Ages_; for if the Number of
Persons of any _Age_ remaining after one Year, be divided by the
difference between that and the number of the _Age_ proposed, it shews
the _Odds_ that there is, that a Person of that _Age_ does not die in a
_Year_. As for Instance, a Person of 25 _Years_ of _Age_ has the Odds of
560 to 7, or 80 to 1, that he does not _die_ in a _Year_: Because that
of 567, living of 25 _Years_ of _Age_, there do die no more than 7 in a
_Year_, leaving 560 of 26 Years old.
So likewise for the _Odds_, that any Person does not die before he
attain any proposed _Age_: Take the _number_ of the remaining Persons of
the _Age_ proposed, and divide it by the difference between it and the
number of those of the _Age_ of the Party proposed; and that shews the
_Odds_ there is between the Chances of the Party's living or dying. As
for Instance; What is the _Odds_ that a Man of 40 lives 7 Years: Take
the number of Persons of 47 Years, which in the Table is 377, and
subtract it from the number of Persons of 40 Years, which is 445, and
the _difference_ is 68: Which shews that the _Persons dying_ in that 7
Years, are 68, and that it is 377 to 68, or 5½ to 1, that a Man of 40
does live 7 Years. And the like for any other _number_ of _Years_.
_Use_ III. But if it be enquired at what number of _Years_, it is an
even Lay that a Person of any _Age_ shall die, this Table readily
performs it; For if the _number_ of Persons _living_ of the _Age_
proposed, be _halfed_, it will be found by the _Table_ at what Year the
said _Number_ is reduced to half by _Mortality_; and that is the _Age_,
to which it is an even Wager, that a Person of the _Age_ proposed shall
arrive before he _die_. As for Instance; A Person of 30 Years of _Age_
is proposed, the number of that _Age_ is 531, the half thereof is 265,
which number I find to be between 57 and 58 Years; so that a Man of 30
may reasonably expect to live between 27 and 28 Years.
_Use_ IV. By what has been said, the _Price_ of _Insurance_ upon Lives
ought to be regulated, and the difference is discovered between the
_Price_ of insuring the _Life_ of a _Man_ of 20 and 50. For Example; It
being 100 to 1, that a Man of 20 dies not in a Year, and but 38 to 1,
for a Man of 50 Years of Age.
_Use_ V. On this depends the Valuation of _Annuities_ upon _Lives_; for
it is plain, that the _Purchaser_ ought to pay for only such a part of
the Value of the _Annuity_, as he has Chances that he is living; and
this ought to be computed yearly, and the Sum of all those yearly Values
being added together, will amount to the Value of the _Annuity_ for the
_Life_ of the Person proposed. Now the present Value of Money payable
after a Term of Years, at any given Rate of Interest, either may be had
from Tables already computed; or almost as compendiously, by the Table
of Logarithms: For the Arithmetical Complement of the Logarithm of
Unity, and its yearly Interest, (that is, of 1,06 for Six _per Cent._
being 9,974694.) being multiplied by the number of Years proposed, gives
the present Value of One Pound payable after the end of so many Years.
Then by the foregoing Proposition, it will be as the number of Persons
living after that Term of Years, to the number dead; so are the Odds
that any one Person is alive or dead. And by consequence, as the Sum of
both, or the number of Persons living of the _Age_ first proposed, to
the number remaining after so many Years, (both given by the Table) so
the present Value of the yearly Sum payable after the Term proposed, to
the Sum which ought to be paid for the Chance the Person has to enjoy
such an _Annuity_ after so many Years. And this being repeated for every
Year of the Person's Life, the Sum of all the present Values of those
Chances is the true Value of the Annuity. This will without doubt appear
to be a most laborious Calculation; but it being one of the principal
Uses of this Speculation, and having found some _Compendia_ for the
Work, I took the pains to compute the following Table, being the short
Result of a not ordinary number of Arithmetical Operations: It shews the
Value of Annuities for every Fifth Year of Age, to the Seventieth, as
follows.
+------+------------+------+------------+------+------------+
| Age. | Years Pur. | Age. | Years Pur. | Age. | Years Pur. |
+------+------------+------+------------+------+------------+
| 1 | 10,28 | 25 | 12,27 | 50 | 9,21 |
| 5 | 13,40 | 30 | 11,72 | 55 | 8,51 |
| 10 | 13,44 | 35 | 11,12 | 60 | 7,60 |
| 15 | 13,33 | 40 | 10,57 | 65 | 6,54 |
| 20 | 12,78 | 45 | 9,91 | 70 | 5,32 |
+------+------------+------+------------+------+------------+
This shews the great Advantage of putting Money into the present _Fund_
lately granted to Their Majesties, giving 14 _per Cent. per Annum_, or
at the Rate of 7 Years Purchase for a Life; when young Lives, at the
usual Rate of Interest, are worth above 13 Years Purchase. It shews
likewise the Advantage of young Lives over those in Years; a Life of Ten
Years being almost worth 13½ Years Purchase, whereas one of 36 is
worth but 11.
_Use_ VI. Two Lives are likewise valuable by the same Rule; for the
number of Chances of each single Life, found in the Table, being
multiplied together, become the Chances of the Two Lives. And after any
certain Term of Years, the Product of the two remaining Sums is the
Chances that both the Persons are living. The Product of the two
Differences, being the numbers of the Dead of both Ages, are the Chances
that both the Persons are dead. And the two Products of the remaining
Sums of the one Age multiplied by those dead of the other, shew the
Chances that there are, that each Party survives the other: Whence is
derived the Rule to estimate the Value of the Remainder of one Life
after another. Now as the Product of the Two Numbers in the Table for
the Two Ages proposed, is to the difference between that Product, and
the Product of the two numbers of Persons deceased in any space of time;
so is the Value of a Sum of Money to be paid after so much time, to the
Value thereof under the Contingency of Mortality. And as the aforesaid
Product of the two Numbers answering to the Ages proposed, to the
Product of the Deceased of one Age multiplied by those remaining alive
of the other; so the Value of a Sum of Money to be paid after any time
proposed, to the Value of the Chances, that the one Party has that he
survives the other, whose number of Deceased you made use of, in the
second Term of the Proportion. This perhaps may be better understood, by
putting _N_ for the number of the younger Age, and _n_ for that of the
Elder; _Y_, _y_ the Deceased of both Ages respectively, and _R_, _r_ for
the Remainders; and _R + Y_ = _N_, and _r + y_ = _n_. Then shall _Nn_ be
the whole Number of Chances; _Nn - Yy_ be the Chances that one of the
two Persons is living, _Yy_ the Chances that they are both dead; _Ry_
the Chances that the elder Person is dead, and the younger living; and
_rY_ the Chances, that the elder is living, and the younger dead. Thus
two Persons of 18 and 35 are proposed, and after 8 Years these Chances
are required. The Numbers for 18 and 35, are 610 and 490; and there are
50 of the First Age dead in 8 Years, and 73 of the Elder Age. There are
in all 610 × 490, or 298900 Chances; of these there are 50 × 73, or
3650, that they are both dead. And as 298900, to 298900 - 3650, or
295250: So is the present Value of a Sum of Money to be paid after 8
Years, to the present Value of a Sum to be paid, if either of the two
live. And as 560 × 73, so are the Chances that the Elder is dead,
leaving the Younger; and as 417 × 50, so are the Chances that the
Younger is dead, leaving the Elder. Wherefore as 610 × 490 to 560 × 73,
so is the present Value of a Sum to be paid at 8 Years end, to the Sum
to be paid for the Chance of the Younger's Survivance; and as 610 × 490
to 417 × 50, so is the same present Value to the Sum to be paid for the
Chance of the Elder's Survivance.
This possibly may be yet better explained, by expounding these Products
by Rectangular Parallelograms, as in _Fig. 7._ wherein _AB_ or _CD_
represents the number of Persons of the younger Age, and _DE_, _BH_
those remaining alive after a certain Term of Years; whence _CE_ will
answer the number of those dead in that time: So _AC_, _BD_ may
represent the number of the elder Age; _AF_, _BI_ the Survivors after
the same Term; and _CF_, _DI_, those of that Age that are dead at that
time. Then shall the whole Parallelogram _ABCD_ be _Nn_, or the Product
of the two Numbers of Persons, representing such a number of Persons of
the two Ages given; and by what was said before, after the Term
proposed, the Rectangle _HD_ shall be as the number of Persons of the
younger Age that survive, and the Rectangle _AE_ as the number of those
that die. So likewise the Rectangles _AI_, _FD_ shall be as the Numbers,
living and dead, of the other Age. Hence the Rectangle _HI_ shall be as
an equal number of both Ages surviving. The Rectangle _FE_ being the
Product of the Deceased, or _Yy_, an equal number of both dead. The
Rectangle _GD_ or _Ry_, a number living of the younger Age, and dead of
the elder: And the Rectangle _AG_ or _rY_ a number living of the elder
Age, but dead of the younger. This being understood, it is obvious, that
as the whole Rectangle _AD_ or _Nn_ is to the _Gnomon FABDEG_ or
_Nn - Yy_, so is the whole number of Persons or Chances, to the number of
Chances that one of the two Persons is living: And as _AD_ or _Nn_ is to
_FE_ or _Yy_, so are all the Chances, to the Chances that both are dead;
whereby may be computed the Value of the Reversion after both Lives. And
as _AD_ to _GD_ or _Ry_, so the whole number of Chances, to the Chances
that the younger is living, and the other dead; whereby may be cast up
what Value ought to be paid for the Reversion of one Life after another,
as in the Case of providing for Clergy-men's Widows, and others, by such
Reversions. And as _AD_ to _AG_, or _rY_, so are all the Chances, to
those that the elder survives the younger. I have been the more
particular, and perhaps tedious, in this Matter, because it is the Key
to the Case of Three Lives, which of it self would not have been so
easie to comprehend.
VII. If Three Lives are proposed, to find the Value of an Annuity during
the continuance of any of those three Lives; the Rule is, _As the
Product of the continual Multiplication of the Three Numbers, in the
Table, answering to the Ages proposed, is to the difference of that
Product, and of the Product of the Three Numbers of the Deceased of
those Ages, in any given Term of Years: So is the present Value of a Sum
of Money, to be paid certainly after so many Years, to the present Value
of the same Sum to be paid, provided one of those Three Persons be
living at the Expiration of that Term._ Which Proportion being yearly
repeated, the Sum of all those present Values will be the Value of an
Annuity granted for three such Lives. But to explain this, together with
all the Cases of Survivance in Three Lives: Let _N_ be the Number in the
Table for the younger Age, _n_ for the second, and ν for the elder
Age; let _Y_ be those dead of the younger Age in the Term proposed, _y_
those dead of the second Age, and υ those of the elder Age; and let
_R_ be the Remainder of the younger Age, _r_ that of the middle Age, and
ρ the Remainder of the elder Age. Then shall _R + Y_ be equal to _N_,
_r + y_ to _n_, and ρ + υ to ν, and the continual Product of the
three Numbers _N_, _n_, ν, shall be equal to the continual Product of
_R + Y × r + y × ρ + υ_, which being the whole Number of Chances for
three Lives, is compounded of the eight Products following. (1) _Rrρ_,
which is the Number of Chances that all three of the Persons are living.
(2) _rρY_, which is the Number of Chances that the two elder Persons
are living, and the younger dead. (3) _Rρy_ the Number of Chances that
the middle Age is dead, and the younger and elder living. (4) _Rrυ_
being the Chances that the two younger are living, and the elder dead.
(5) _ρYy_ the Chances that the two younger are dead, and the elder
living. (6) _rYυ_ the Chances that the younger and elder are dead, and
the middle Age living. (7) _Ryυ_, which are the Chances that the
younger is living, and the two other dead. And Lastly and Eighthly,
_Yyυ_, which are the Chances that all three are dead. Which latter
subtracted from the whole Number of Chances _Nnν_, leaves _Nnν -
Yyυ_ the Sum of all the other seven Products; in all of which one or
more of the three Persons are surviving.
To make this yet more evident, I have added _Fig. 8._ wherein these
eight several Products are at one view exhibited. Let the rectangled
Parallelepipedon _ABCDEFGH_ be constituted of the sides _AB_, _GH_,
_&c._ proportional to _N_ the Number of the younger Age; _AC_, _BD_,
_&c._ proportional to _n_; and _AG_, _CE_, _&c._ proportional to the
Number of the elder, or ν. And the whole Parallelepipedon shall be as
the Product _Nnν_, or our whole Number of Chances. Let _BP_ be as _R_,
and _AP_ as _Y_; let _CL_ be as _r_, and _Ln_ as _y_; and _GN_ as ρ,
and _NA_ as υ; and let the Plain _PRea_ be made parallel to the Plain
_ACGE_; the Plain _NVbY_ parallel to _ABCD_; and the Plain _LXTQ_
parallel to the Plain _ABGH_. And our first Product _Rrρ_ shall be as
the Solid _STWIFZeb_. The Second, or _rρY_ will be as the Solid
_EYZeQSMI_. The Third, _Rρy_, as the Solid _RHOVWIST_. And the Fourth,
_Rrυ_, as the Solid _ZabDWXIK_. Fifthly, _ρYy_, as the Solid
_GQRSIMNO_. Sixthly, _rYυ_, as _IKLMGYZA_. Seventhly, _Ryυ_, as the
Solid _IKPOBXVW_. And Lastly, _AIKLMNOP_ will be as the Product of the 3
Numbers of Persons dead, or _Yyυ_. I shall not apply this in all the
Cases thereof, for brevity sake; only to shew in one how all the rest
may be performed, let it be demanded what is the Value of the Reversion
of the younger Life after the two elder proposed. The proportion is as
the whole Number of Chances, or _Nnν_ to the Product _Ryυ_; so is
the certain present Value of the Sum payable after any Term proposed, to
the Value due to such Chances as the younger Person has to bury both the
elder, by the Term proposed; which therefore he is to pay for. Here it
is to be noted, that the first Term of all these Proportions is the same
throughout, _viz._ _Nnν_. The second changing yearly according to the
Decrease of _R_, _r_, _ρ_, and Increase of _Y_, _y_, _υ_. And the
third are successively the present Values of Money payable after one,
two, three, _&c._ years, according to the Rate of Interest agreed on.
These Numbers, which are in all Cases of Annuities of necessary Use, I
have put into the following Table, they being Decimal Values of one
Pound payable after the Number of Years in the Margent, at the Rate of 6
_per Cent._
+--------+-------------+--------+-------------+--------+-------------+
| Years. | Pres. Value | Years. | Pres. Value | Years. | Pres. Value |
| | of 1 _l._ | | of 1 _l._ | | of 1 _l._ |
+--------+-------------+--------+-------------+--------+-------------+
| 1 | 0,9434 | 19 | 0,3305 | 37 | 0,1158 |
| 2 | 0,8900 | 20 | 0,3118 | 38 | 0,1092 |
| 3 | 0,8396 | 21 | 0,2941 | 39 | 0,1031 |
| 4 | 0,7921 | 22 | 0,2775 | 40 | 0,0972 |
| 5 | 0,7473 | 23 | 0,2618 | 45 | 0,0726 |
| 6 | 0,7050 | 24 | 0,2470 | 50 | 0,0543 |
+--------+-------------+--------+-------------+--------+-------------+
| 7 | 0,6650 | 25 | 0,2330 | 55 | 0,0406 |
| 8 | 0,6274 | 26 | 0,2198 | 60 | 0,0303 |
| 9 | 0,5919 | 27 | 0,2074 | 65 | 0,0227 |
| 10 | 0,5584 | 28 | 0,1956 | 70 | 0,0169 |
| 11 | 0,5268 | 29 | 0,1845 | 75 | 0,0126 |
| 12 | 0,4970 | 30 | 0,1741 | 80 | 0,0094 |
+--------+-------------+--------+-------------+--------+-------------+
| 13 | 0,4688 | 31 | 0,1643 | 85 | 0,0071 |
| 14 | 0,4423 | 32 | 0,1550 | 90 | 0,0053 |
| 15 | 0,4173 | 33 | 0,1462 | 95 | 0,0039 |
| 16 | 0,3936 | 34 | 0,1379 | 100 | 0,0029 |
| 17 | 0,3714 | 35 | 0,1301 | | |
| 18 | 0,3503 | 36 | 0,1227 | | |
+--------+-------------+--------+-------------+--------+-------------+
It were needless to advertise, that the great trouble of working so many
Proportions will be very much alleviated by using Logarithms; and that
instead of using _Nnν - Yyυ_ for the second Term of the Proportion
in finding the Value of Three Lives, it may suffice to use only _Yyυ_,
and then deducting the fourth Term so found out of the third, the
Remainder shall be the present Value sought; or all these fourth Terms
being added together, and deducted out of the Value of the certain
Annuity for so many Years, will leave the Value of the contingent
Annuity upon the Chance of Mortality of all those Three Lives. For
Example; Let there be Three Lives of 10, 30, and 40 Years of Age
proposed, and the Proportions will be thus;
As 661 in 531 in 445 or 156190995, or _Nnν_
to 8 in 8 in 9, or 576, or _Yyυ_ for the first
Year, so 0,9434. to 0,00000348.
To 15 in 16 in 18, or 4320, for the second
Year, so 0,8900. to 0,00002462.
To 21 in 24 in 28, or 14112 for the third
Year, so 0,8396. to 0,00008128.
To 27 in 32 in 38, for the fourth Year, so
0,7921. to 0,00016650.
To 33 in 41 in 48, for the fifth Year, so
0,7473. to 0,00031071.
To 39 in 50 in 58, for the sixth Year, so
0,7050. to 0,00051051.
And so forth to the 60th Year, when we suppose the elder Life of Forty
certainly to be expired; from whence till Seventy we must compute for
the First and Second only, and from thence to Ninety for the single
youngest Life. Then the Sum Total of all these Fourth Proportionals
being taken out of the Value of a certain Annuity for 90 Years, being
16,58 Years Purchase, shall leave the just Value to be paid for an
Annuity during the whole Term of the Lives of Three Persons of the Ages
proposed. And note, that it will not be necessary to compute for every
Year singly; but that in most Cases every 4th or 5th Year may suffice,
interpoling for the intermediate Years _seceundum artem_.
It may be objected, that the different _Salubrity_ of Places does hinder
this Proposal from being _universal_; nor can it be denied. But by the
Number that die, being 1174. _per Annum_ in 34000, it does appear that
about a 30th part die yearly, as Sir _William Petty_ has computed for
_London_; and the Number that die in Infancy, is a good Argument that
the Air is but indifferently salubrious. So that by what I can learn,
there cannot perhaps be one better Place proposed for a Standard. At
least 'tis desired, that in Imitation hereof the Curious in other Cities
would attempt something of the same Nature, than which nothing perhaps
can be more useful.
Were this _Calculus_ founded on the Experience of a very great number of
Years, it would be very well worth the while to think of Methods for
facilitating the Computation of the Value of two, three, or more Lives;
which, as proposed in my former, seems (as I am inform'd) a Work of too
much Difficulty for the ordinary Arithmetician to undertake.
I have sought, If it were possible, to find a Theorem that might be more
concise than the Rules there laid down, but in vain; for all that can be
done to expedite it, is, by Tables of Logarithms ready computed, to
exhibit the _Rationes_ of _N_ to _Y_ in each single Life, for every
third, fourth, or fifth Year of Age, as occasion shall require; and
these Logarithms being added to the Logarithms of the present Value of
Money payable after so many Years, will give a Series of Numbers, the
Sum of which will shew the Value of the Annuity sought. However, for
each Number of this Series, two Logarithms for a single Life, three for
two Lives, and four for three Lives, must necessarily be added together.
If you think the Matter, under the Uncertainties I have mentioned, to
deserve it, I shall shortly give you such a Table of Logarithms, as I
speak of, and an Example or two of the use thereof: But by Vulgar
Arithmetick, the Labour of these Numbers were immense; and nothing will
more recommend the useful Invention of Logarithms to all Lovers of
Numbers, than the advantage of Dispatch in this and such like
Computations.
Besides the Uses mentioned, it may perhaps not be an unacceptable thing
to infer from the same Tables, how unjustly we repine at the shortness
of our Lives, and think our selves wronged if we attain not old Age;
whereas it appears hereby, that the one half of those that are born are
dead in Seventeen Years time, 1238 being in that time reduced to 616. So
that instead of murmuring at what we call an untimely Death, we ought
with Patience and Unconcern to submit to that Dissolution which is the
necessary Condition of our perishable Materials, and of our nice and
frail Structure and Composition: And to account it as a Blessing that we
have survived, perhaps by many Years, that Period of Life, whereat the
one half of the whole Race of Mankind does not arrive.
A second Observation I make upon the said Table, is that the Growth and
Increase of Mankind is not so much stinted by any thing in the Nature of
the _Species_, as it is from the cautious difficulty most People make to
adventure on the State of _Marriage_, from the Prospect of the Trouble
and Charge of providing for a Family. Nor are the poorer sort of People
herein to be blamed, since their difficulty of subsisting is occasion'd
by the unequal Distribution of Possessions, all being necessarily fed
from the Earth, of which yet so few are Masters. So that besides
themselves and Families, they are yet to work for those who own the
Ground that feeds them: And of such does by very much the greater part
of Mankind consist; otherwise it is plain, that there might well be four
times as many Births as we now find. For by Computation from the Table,
I find that there are nearly 15000 Persons above 16, and under 45, of
which at least 7000 are Women capable to bear Children. Of these
notwithstanding there are but 1238 born yearly, which is but little more
than a sixth part: So that about one in six of these Women do breed
yearly; whereas were they all married, it would not appear strange or
unlikely, that four of six should bring a Child every Year. The
Political Consequences hereof I shall not insist on; only the Strength
and Glory of a King being in the multitude of his Subjects, I shall only
hint, that above all things, Celibacy ought to be discouraged, as, by
extraordinary Taxing and Military Service: And those who have numerous
Families of Children to be countenanced and encouraged by such Laws as
the _Jus trium Liberorum_ among the _Romans_. But especially, by an
effectual Care to provide for the Subsistence of the Poor, by finding
them Employments, whereby they may earn their Bread, without being
chargeable to the Publick.
_A Discourse concerning _Gravity_, and its Properties, wherein the
Descent of _Heavy Bodies_, and the Motion of _Projects_ is briefly,
but fully handled: Together with the _Solution_ of a _Problem_ of
great Use in _Gunnery_. By _E. Halley_._
Nature, amidst the great Variety of _Problems_, wherewith She exercises
the Wits of Philosophical Men, scarce affords any one wherein the Effect
is more visible, and the Cause more concealed, than in those of the
_Phænomena_ of _Gravity_. Before we can go alone, we must learn to
defend our selves from the Violence of its Impulse, by not trusting the
_Center_ of _Gravity_ of our Bodies beyond our reach; and yet the
acutest Philosophers, and the subtilest Enquirers into the Original of
this Motion, have been so far from satisfying their Readers, that they
themselves seem little to have understood the Consequences of their own
_Hypotheses_.
_Des Cartes_ his Notion, I must needs confess to be to me incomprehensible,
while he will have the Particles of his _Cœlestial Matter_, by being
reflected on the Surface of the _Earth_, and so ascending therefrom, to
drive down into their Places those _Terrestrial Bodies_ they find above
them: This is, as near as I can gather, the Scope of the 20, 21, 22, and
23 _Sections_ of the last Book of his _Principia Philosophiæ_; yet
neither he, nor any of his Followers, can shew how a Body suspended in
_Libero Æthere_, shall be carried downwards by a continual Impulse
tending upwards, and acting upon all its Parts equally: And besides the
Obscurity wherewith he expresses himself, particularly, _Sect. 23._ does
sufficiently argue according to his own Rules, the confused _Idea_ he
had of the thing he wrote.
Others, and among them Dr. _Vossius_, assert the Cause of the _Descent_
of _heavy Bodies_, to be the _Diurnal Rotation_ of the _Earth_ upon its
_Axis_, without considering, that according to the Doctrine of Motion
fortified with Demonstration, all Bodies moved _in Circulo_, would
recede from the Center of their Motion; whereby the contrary to
_Gravity_ would follow, and all loose Bodies would be cast into the Air
in a _Tangent_ to the _Parallel_ of _Latitude_, without the intervention
of some other Principle to keep them fast, such as is that of _Gravity_.
Besides, the Effect of this Principle is throughout the whole Surface of
the Globe found nearly equal; and certain Experiments have proved it
rather less near the _Æquinoctial_, than towards the _Poles_; which
could not be by any means, if the _Diurnal Rotation_ of the _Earth_ upon
its _Axis_ were the Cause of _Gravity_; for where the Motion was
swiftest, the Effect would be most considerable.
Others assign the Pressure of the _Atmosphere_, to be the Cause of this
Tendency towards the Center of the Earth; but unhappily they have
mistaken the Cause for the Effect; it being from undoubted Principles
plain, that the _Atmosphere_ has no other Pressure but what it derives
from its _Gravity_; and that the Weight of the upper Parts of the _Air_,
pressing on the lower Parts thereof, do so far bend the Springs of that
_Elastick_ Body, as to give it a Force equal to the Weight that
compress'd it, having of it self no force at all: And supposing it had,
it will be very hard to explain the _Modus_, how that Pressure should
occasion the Descent of a Body circumscribed by it, and pressed equally
above and below, without some other Force to draw, or thrust it
downwards. But to demonstrate the contrary of this Opinion, an
_Experiment_ was long since shewn before the _Royal Society_, whereby it
appeared, that the _Atmosphere_ was so far from being the Cause of
_Gravity_, that the Effects thereof were much more vigorous, where the
Pressure of the _Atmosphere_ was taken off; for a long _Glass-Receiver_
having a light Down-feather included, being evacuated of Air, the
Feather, which in the Air would hardly sink, did _in vacuo_ descend with
nearly the same _Velocity_, as if it had been a Stone.
Some think to illustrate this Descent of Heavy Bodies, by comparing it
with the Vertue of the _Loadstone_; but setting aside the difference
there is in the manner of their Attractions, the _Loadstone_ drawing
only in and about its Poles, and the Earth near equally in all Parts of
its Surface, this Comparison avails no more than to explain _ignotum per
æque ignotum_.
Others assign a certain _Sympæthetical Attraction_ between the Earth and
its Parts, whereby they have, as it were, a desire to be united, to be
the Cause we enquire after: But this is so far from explaining the
_Modus_, that it is little more, than to tell us in other Terms, that
Heavy Bodies descend, because they descend.
This, I say, not that I can pretend to substitute any Solution of this
Important Philosophical Problem, that shall more happily explicate the
Appearances of Gravity; only it may be serviceable to those with whom
the Credit of great Authors sways much, and who too readily assent _in
Verba Magistri_, to let them see that their Books are not always
infallible: Besides, the detection of Errors is the first and surest
Step towards the discovery of Truth.
Though the efficient Cause of _Gravity_ be so obscure, yet the final
Cause thereof is clear enough; for it is by this single _Principle_,
that the _Earth_ and all the _Cœlestial Bodies_ are kept from
_Dissolution_; the least of their _Particles_ not being suffer'd to
recede far from their _Surfaces_, without being immediately brought down
again by Virtue of this _Natural Tendency_; which, for their
Preservation, the Infinite Wisdom of their _Creator_ has ordained to be
towards each of their _Centers_; nor can the _Globes_ of the _Sun_ and
_Planets_ otherwise be destroy'd, but by taking from them this Power of
keeping their Parts united.
The Affections or Properties of _Gravity_, and its manner of acting upon
_Bodies falling_, have been in a great measure discovered, and most of
them made out by _Mathematical Demonstration_ in this our _Century_, by
the accurate diligence of _Galilæus_, _Torricellius_, _Hugenius_, and
others, and now lately by our worthy Countryman, Mr. _Isaac Newton_,
which Properties it may be very material here to enumerate, that they
may serve for a Foundation to all those that shall be willing to spend
their Thoughts in search of the true Cause of this _Descent of Bodies_.
The first Property is, That by this Principle of _Gravitation_, all
Bodies do descend towards a Point, which either is, or else is very near
to the Center of Magnitude of the Earth and Sea, about which the Sea
forms it self exactly into a _Spherical Surface_, and the _Prominences_
of the Land, considering the Bulk of the whole, differ but insensibly
therefrom.
_Secondly_, That this Point or Center of _Gravitation_, is fix'd within
the _Earth_, or at least has been so, ever since we have any _Authentick
History_: For a Consequence of its Change, though never so little, would
be the over-flowing of the low Lands on that side of the _Globe_ towards
which it approached, and the leaving new Islands bare on the opposite
side, from which it receded; but for this Two Thousand Years it appears,
that the low Islands of the _Mediterranean Sea_ (near to which the
ancientest Writers liv'd) have continued much at the same height above
the Water, as they now are found; and no _Inundations_ or _Recesses_ of
the _Sea_ arguing any such Change, are recorded in History; excepting
the _Universal Deluge_, which can no better way be accounted for, than
by supposing this Center of _Gravitation_ removed for a time, towards
the middle of the then inhabited Parts of the World; and a change of its
Place, but the Two Thousandth Part of the _Radius_ of this _Globe_, were
sufficient to bury the Tops of the highest Hills under Water.
_Thirdly_, That in all Parts of the _Surface_ of the _Earth_, or rather
in all Points equidistant from its _Center_, the Force of _Gravity_ is
nearly equal; so that the length of the _Pendulum_ vibrating _Seconds of
Time_, is found in all Parts of the World to be very near the same. 'Tis
true at St. _Helena_, in the _Latitude_ of 16 Degrees _South_, I found
that the _Pendulum_ of my Clock, which vibrated _Seconds_, needed to be
made shorter than it had been in _England_, by a very sensible Space
(but which at that time I neglected to observe accurately) before it
would keep time; and since the like Observations have been made by the
_French Observers_, near the _Æquinoctial_: Yet I dare not affirm, that
in mine it proceeded from any other Cause, than the great Height of my
Place of Observation above the _Surface_ of the _Sea_, whereby the
_Gravity_ being diminished, the length of the _Pendulum_ vibrating
_Seconds_, is proportionably short'ned.
_Fourthly_, That _Gravity_ does equally affect all _Bodies_, without
regard either to their _Matter_, _Bulk_, or _Figure_; so that the
Impediment of the _Medium_ being removed, the most compact and most
loose, the greatest and smallest _Bodies_ would descend the same
_Spaces_ in equal Times; the Truth thereof will appear from the
Experiment I before-cited. In these two last Particulars, is shewn the
great difference between _Gravity_ and _Magnetism_, the one affecting
only _Iron_, and that towards its _Poles_, the other all _Bodies_ alike
in every part. As a _Corollary_, from hence it will follow, that there
is no such thing as _positive Levity_, those things that appear light,
being only comparatively so; and whereas several things rise and swim in
_Fluids_, 'tis because, Bulk for Bulk, they are not so heavy as those
_Fluids_; nor is there any Reason why _Cork_, for Instance, should be
said to be light, because it swims on Water, any more than _Iron_,
because it swims on _Mercury_.
_Fifthly_, That this Power increases as you descend, and decreases as
you ascend from the Center, and that in the Proportion of the Squares of
the _Distances_ therefrom _reciprocally_, so as at a double Distance to
have but a quarter of the Force; this Property is the Principle on which
Mr. _Newton_ has made out all the _Phænomena_ of the _Cœlestial
Motions_, so easily and naturally, that its Truth is past Dispute.
Besides that, it is highly rational, that the _attractive_ or
_gravitating_ Power should exert it self more vigorously in a small
Sphere, and weaker in a greater, in proportion as it is contracted or
expanded; and if so, seeing that the _Surfaces_ of _Spheres_ are as the
_Squares_ of their _Radii_, this Power, at several Distances, will be as
the _Squares_ of those _Distances reciprocally_; and then its whole
Action upon each _Spherical Surface_, be it great or small, will be
always equal. And this is evidently the Rule of _Gravitation_ towards
the _Centers_ of the _Sun_, _Jupiter_, _Saturn_ and the _Earth_, and
thence is reasonably inferred, to be the general Principle observed by
_Nature_, in all the rest of the _Cœlestial Bodies_.
These are the principal Affections of _Gravity_, from which the Rules of
the _Fall_ of _Bodies_, and the _Motion_ of _Projects_ are
_Mathematically_ deducible. Mr. _Isaac Newton_ has shew'd how to define
the Spaces of the _Descent_ of a _Body_, let fall from any given height,
down to the _Center_, supposing the _Gravitation_ to increase, as in the
fifth Property; but considering the smallness of heighth, to which any
_Project_ can be made ascend, and over how little an _Arch_ of the
_Globe_ it can be cast by any of our _Engines_, we may well enough
suppose the _Gravity_ equal throughout, and the Descents of _Projects_
in parallel Lines, which in Truth are towards the _Center_, the
difference being so small as by no means to be discovered in _Practice_.
The _Opposition_ of the _Air_, 'tis true, is considerable against all
light Bodies moving through it, as likewise against small ones (of which
more hereafter) but in great and ponderous Shot, this Impediment is
found by _Experience_ but very small, and may safely be neglected.
_Propositions concerning the Descent of Heavy Bodies, and the Motion of
_Projects_._
_Prop. I._ The _Velocities_ of _Falling Bodies_, are proportionate to
the Times from the beginning of their _Falls_.
This follows, for that the Action of _Gravity_ being _continual_, in
every Space of Time, the falling Body receives a new Impulse, equal to
what it had before, in the same Space of Time, received from the same
Power: For Instance, in the first Second of Time, the falling Body has
acquired a _Velocity_, which in that time would carry it to a certain
Distance, suppose 32 Foot, and were there no new Force, would descend at
that rate with an _equable Motion_: But in the next Second of Time, the
same Power of _Gravity_ continually acting thereon, superadds a new
_Velocity_ equal to the former; so that at the end of two Seconds, the
_Velocity_ is double to what it was at the end of the first, and after
the same manner may it be proved to be triple, at the end of the third
Second, and so on. Wherefore the _Velocities_ of _falling Bodies_, are
proportionate to the Time of their _Falls_, _Q. E. D._
[Illustration: _Plate 4. pag. 310_]
_Prop. II._ The _Spaces_ described by the Fall of a Body, are as the
_Squares_ of the Times, from the beginning of the _Fall_.
_Demonstration._ Let AB (_Fig. 9. Tab. 4._) represent the _Time_ of the
_Fall_ of a _Body_, BC perpendicular to AB, the _Velocity_ acquired at
the end of the _Fall_, and draw the Line AC; then divide the Line AB
representing the Time, into as many equal Parts as you please, as b, b,
b, b, _&c._ and through these Points draw the Lines bc, bc, bc, bc,
_&c._ parallel to BC, 'tis manifest that the several Lines, bc,
represent the several _Velocities_ of the falling Body, in such Parts of
the _Time_ as Ab is of AB, by the former Proposition. It is evident
likewise, that the _Area_ ABC is the Sum of all the Lines bc being
taken, according to the Method of _Indivisibles_, infinitely many; so
that the _Area_ ABC represents the Sum of all the _Velocities_, between
none and BC supposed infinitely many; which Sum is as the Space
descended in the Time represented by AB. And by the same Reason the
_Areas_ Abc, will represent the Spaces descended in the Times Ab; so
then the Spaces descended in the Times AB, Ab, are as the _Areas_ of the
_Triangles_ ABC, Abc, which by the 20th of the 6 of _Euclid_, are as the
_Squares_ of their _Homologous Sides_ AB, Ab, that is to say, of the
_Times_: Wherefore the Descents of _falling Bodies_, are as the
_Squares_ of the Times of their _Fall_, _Q. E. D._
_Prop. III._ The _Velocity_ which a _falling Body_ acquires in any Space
of time, is double to that, wherewith it would have moved the Space,
descended by an equable Motion, in the same _time_.
_Demonstration._ Draw the Line EC parallel to AB, and AE parallel to BC
in the same _Fig. 9._ and compleat the _Parallelogram_ ABCE, it is
evident that the _Area_ thereof may represent the Space, a _Body_ moved
equably with the _Velocity_ BC would describe in the Time AB, and the
_Triangle_ ABC represents the _Space_ describ'd by the _Fall_ of a
_Body_, in the same Time AB, by the second Proposition. Now the
_Triangle_ ABC is half of the _Parallelogram_ ABCE, and consequently the
Space described by the _Fall_, is half what would have been described by
an _equable Motion_ with the _Velocity_ BC, in the same Time; wherefore
the _Velocity_ BC at the end of the _Fall_, is double to that
_Velocity_, which in the Time AB, would have described the _Space
fallen_, represented by the _Triangle_ ABC with an _equable Motion_, _Q.
E. D._
_Prop. IV._ All _Bodies_ on or near the Surface of the _Earth_, in their
_Fall_, descend so, as at the end of the first Second of Time, they have
described 16 Feet, 1 Inch, _London Measure_, and acquired the _Velocity_
of 32 Feet, 2 Inches, in a Second.
This is made out from the 25th Proposition of the second Part of that
excellent Treatise of Mr. _Hugenius de Horologio Oscillatorio_; wherein
he demonstrates the time of the least _Vibrations_ of a _Pendulum_, to
be to the Time of the _Fall_ of a _Body_, from the heighth of half the
length of the _Pendulum_, as the _Circumference_ of a _Circle_ to its
_Diameter_; whence, as a _Corollary_, it follows, That as the _Square_
of the _Diameter_ to the _Square_ of the _Circumference_, so half the
length of the _Pendulum_ vibrating _Seconds_, to the _Space_ described
by the _Fall_ of a _Body_ in a _Second_ of _Time_: And the Length of the
_Pendulum_ vibrating _Seconds_, being found 39, 125, or ⅛ Inches, the
_Descent_ in a _Second_ will be found by the aforesaid _Analogy_ 16 Foot
and 1 Inch; and, by the third Proposition, the _Velocity_ will be double
thereto; and near to this it hath been found by several Experiments,
which by reason of the _swiftness_ of the _Fall_, cannot so exactly
determine its _Quantity_. The Demonstration of _Hugenius_ being the
Conclusion of a long Train of _Consequences_, I shall for brevity sake
omit; and refer you to his Book, where these things are more amply
treated of.
From these Four _Propositions_, all _Questions_ concerning the
_Perpendicular Fall of Bodies_, are easily _solved_, and either _Time_,
_Height_, or _Velocity_ being assign'd, one may readily find the other
two. From them likewise is the Doctrine of _Projects_ deducible,
assuming the two following _Axioms_; _viz._ That a _Body_ set a moving,
will move on continually in a right _Line_ with an _equable Motion_,
unless some other Force or Impediment intervene, whereby it is
accelerated, or retarded, or deflected.
_Secondly_, That a _Body_ being agitated by two _Motions_ at a time,
does by their _compounded Forces_ pass through the same _Points_, as it
would do, were the two _Motions divided_ and acted _successively_. As
for Instance, Suppose a _Body_ moved in the Line GF, (_Fig. 1. Tab. 5._)
from G to R, and there stopping, by another _Impulse_, suppose it moved
in a _Space_ of _Time_ equal to the former, from R towards K, to V. I
say, the _Body_ shall pass through the Point to V, though these two
_several Forces_ acted both in the _same time_.
_Prop._ V. The _Motion_ of all _Projects_ is in the _Curve_ of a
_Parabola_: Let the _Line_ GRF (in _Fig._ 1.) be the _Line_ in which the
_Project_ is directed, and in which by the first _Axiom_ it would move
equal _Spaces_ in equal _Times_, were it not deflected downwards by the
Force of _Gravity_. Let GB be the _Horizontal Line_, and GC a
_Perpendicular_ thereto. Then the _Line_ GRF being divided into equal
Parts, answering to equal _Spaces_ of _Time_, let the _Descents_ of the
_Project_ be laid down in _Lines parallel_ to GC, proportioned as the
_Squares_ of the _Lines_ GS, GR, GL, GF, or as the _Squares_ of the
_Times_, from S to T, from R to V, from L to X, and from F to B, and
draw the _Lines_ TH, VD, XY, BC parallel to GF; I say, the Points T, V,
X, B, are Points in the _Curve_ described by the _Project_, and that
that _Curve_ is a _Parabola_. By the second _Axiom_, they are Points in
the _Curve_; and the Parts of the _Descent_ GH, GD, GY, GC, = to ST, RV,
LX, FB, being as the _Squares_ of the _Times_ (by the _Second
Proposition_) that is, as the _Squares_ of the _Ordinates_, HT, DU, YX,
BC, equal to GS, GR, GL, GF, the _Spaces_ measured in those Times; and
there being no other _Curve_ but the _Parabola_, whose Parts of the
_Diameter_ are as the _Squares_ of the _Ordinates_, it follows that the
_Curve_ describ'd by a _Project_, can be no other than a _Parabola_: And
saying, as RU the _Descent_ in any _time_, to GR or UD the _direct
Motion_ in the same _time_, so is UD to a _third_ proportional; that
_third_ will be the _Line_ call'd by all Writers of _Conicks_, the
_Parameter_ of the _Parabola_ to the _Diameter_ GC, which is always the
same in _Projects_ cast with the same _Velocity_: And the _Velocity_
being defined by the Number of _Feet_ moved in a _Second_ of Time, the
_Parameter_ will be found by dividing the _Square_ of the _Velocity_, by
16 _Feet_, 1 _Inch_, the _Fall_ of a _Body_ in the same _Time_.
_Lemma._
The _Sine_ of the double of any _Arch_, is equal to twice the _Sine_ of
that _Arch_ into its _Co-sine_, divided by _Radius_; and the _versed
Sine_ of the _double_ of any _Arch_ is equal to twice the _Square_ of
the _Sine_ thereof divided by _Radius_.
Let the _Arch_ BC (in _Fig. 2. Tab. 5._) be double the _Arch_ BF, and A
the _Center_; draw the _Radii_ AB, AF, AC, and the _Chord_ BDC, and let
fall BE perpendicular to AC, and the _Angle_ EBC, will be equal to the
_Angle_ ABD, and the _Triangle_ BCE, will be like to the _Triangle_ BDA;
wherefore it will be as AB to AD, so BC or twice BD, to BE; that is, as
_Radius_ to _Co-sine_, so twice _Sine_ to _Sine_ of the double _Arch_.
And as AB to BD, so twice BD or BC to EC, that is, as _Radius_ to
_Sine_, so twice that _Sine_, to the _Versed Sine_ of the double _Arch_;
which two _Analogies_ resolved into _Equations_, are the _Propositions_
contained in the _Lemma_ to be proved.
_Prop._ VI. The _Horizontal_ Distances of _Projections_ made with the
same _Velocity_, at several _Elevations_ of the _Line_ of Direction, are
as the _Sines_ of the doubled _Angles_ of _Elevation_.
Let GB (_Fig._ 1) the _Horizontal_ Distance be = _z_, the _Sine_ of the
_Angle_ of _Elevation_, FGB, be = _s_, its _Co-sine_ = _c_, _Radius_ =
_r_, and the _Parameter_ = _p_. It will be as _c_ to _s_; so _z_ to
_sz_/_c_ = FB = GC, and by reason of the _Parabola_ _psz_/_c_ = to the
_Square_ of CB, or GF; Now as _c_ to _r_, so is _z_ to _zr_/_c_ = GF,
and its _Square_ _zzrr_/_cc_ will be therefore = to _psz_/_c_: Which
_Equation_ reduced will be _psc_/_rr_ = _z_. But by the former _Lemma_
2_sc_/_r_ is equal to the _Sine_ of the double _Angle_, whereof _s_ is
the _Sine_: Wherefore 'twill be as _Radius_ to _Sine_ of double the
_Angle_ FGB, so is half the _Parameter_, to the _Horizontal Range_ or
_Distance_ sought; and at the several _Elevations_, the _Ranges_ are as
the _Sines_ of the double _Angles_ of _Elevation_, _Q. E. D._
_Corollary._
Hence it follows, that half the _Parameter_ is the greatest _Randon_,
and that that happens at the _Elevation_ of 45 Degrees, the _Sine_ of
whose double is _Radius_. Likewise that the _Ranges_ equally distant
above and below 45 are equal, as are the _Sines_ of all double _Arches_,
to the _Sines_ of their doubled _Complements_.
_Prop._ VII. The _Altitudes_ of _Projections_ made with the same
_Velocity_, at several _Elevations_, are as the _versed Sines_ of the
doubled _Angles_ of _Elevation_: As _c_ is to _s_; so is _psc/rr_ = GB
to _pss/rr_ = BF: and UK = RU = BF/4, the _Altitude_ of the _Projection_
= _psc/4rr_. Now by the foregoing _Lemma_ _2ss/r_ = to the _versed Sine_
of the double _Angle_, and therefore it will be as _Radius_, to _versed
Sine_ of double the _Angle_ FGB, so an 8th of the _Parameter_ to the
height of the _Projection_ VK; and so these heights at several
_Elevations_, are as the said _versed Sines_, _Q. E. D._
_Corollary._
From hence it is plain, that the greatest _Altitude_ of the
perpendicular _Projection_ is a 4th of _Parameter_, or half the greatest
_Horizontal Range_; the _versed Sine_ of 180 Degrees being = _2r_.
_Prop._ VIII. The _Lines_ GF, or Times of the Flight of a _Project_ cast
with the same Degree of _Velocity_ at different _Elevations_, are as the
_Sines_ of the _Elevations_.
As _c_ is to _r_; so is _psc/rr_ = GB by the 6 Prop. to _ps/r_ GF; that
is, as _Radius_ to _Sine_ of _Elevation_, so the _Parameter_ to the
_Line_ GF; so the _Lines_ GF are as the _Sines_ of _Elevation_, and the
_Times_ are proportional to those _Lines_; wherefore the _Times_ are as
the _Sines_ of _Elevation_: _Ergo constat propositio_.
_Prop._ IX. Problem. A _Projection_ being made as you please, having the
Distance and Altitude, or Descent, of an Object, through which the
Project passes, together with the _Angle_ of _Elevation_ of the _Line_
of _Direction_; to find the _Parameter_ and _Velocity_, that is (in
_Fig._ 1.) having the _Angle_ FGB, GM, and MX.
_Solution._ As _Radius_ to _Secant_ of FGB, so GM the _Distance_ given
to GL; and as _Radius_ to _Tangent_ of FGB, so GM to LM. Then LM - MX in
_Heights_, or + MX in _Descents_; or else MX - ML, if the _Direction_ be
below the _Horizontal Line_, is the _Fall_ in the _Time_ that the direct
_Impulse_ given in G would have carried the Project from G to L = LX =
GY; then by Reason of the _Parabola_, as LX or GY, is to GL or YX, so is
GL to the _Parameter_ sought. To find the _Velocity_ of the _Impulse_:
by Prop. 2, and 4, find the Time in Seconds that a Body would fall the
Space LX; and by that dividing the Line GL, the _Quote_ will be the
_Velocity_, or Space moved in a Second sought, which is always a mean
Proportional between the _Parameter_, and 16 Feet, 1 Inch.
_Prop_. X. Problem 2. Having the _Parameter_, Horizontal Distance, and
Height or Descent of an _Object_, to find the Elevations of the Line of
Direction necessary to hit the given _Object_; that is, having GM, MX,
and the greatest _Randon_ equal to half the _Parameter_; to find the
_Angles_ FGB.
Let the _Tangent_ of the _Angle_ sought be = _t_, the _Horizontal
Distance_ GM = _b_, the Altitude of the _Object_ MX = _h_, the
_Parameter_ = _p_, and _Radius_ = _r_, and it will be,
As _r_ to _t_, so _b_ to _tb/r_ = ML and
_tb/r ∓ h_ {in ascents|in descents} = LX, and
_ptb/r ∓ ph_ = GL _quad._ = XY _quad. ratione Parabolæ_; but
_bb ∓ ttbb/rr_ = GL _quad._ 47. 1. _Euclid_. Wherefore
_ptb/r ∓ ph_ = _bb ∓ ttbb/rr_ which Equation transposed, is
_ttbb/rr_ = _ptb/r ∓ ph - bb_, divided by _bb_ is
_tt/rr_ = _pt/br ∓ ph/bb_ - 1.
this Equation shews the Question to have 2 Answers, and the Roots
thereof are
_t/r_ = _p/2h_ ∓ √((_pp ∓ 4ph_)/_4bb_) - 1;
from which I derive the following Rule.
Divide half the _Parameter_ by the Horizontal distance, and keep the
Quote; _viz._ _p/2b_ then say, as _square_ of the _distance_ given to
the half _Parameter_, so half _Parameter_ ∓ double {height|descent} to
the _square_ of a _Secant_ = (_pp ∓ 4ph_)/(_4bb_). The _Tangent_
answering to that _Secant_, will be
√((_pp ∓ 4ph_)/4_bb_) - 1 or Square of Radius,
so then the sum and difference of the afore-found _Quote_, and this
_Tangent_ will be the Roots of the _Equation_, and the _Tangents_ of the
_Elevations_ sought.
Note here, that in _Descents_, if the _Tangent_ exceed the _Quote_, as
it does when _ph_ is more than _bb_, the _direction_ of the lower
_Elevation_ will be below the _Horizon_, and if _ph_ = _bb_, it must be
directed _Horizontal_, and the _Tangent_ of the upper _Elevation_ will
be _pr/b_: Note likewise, that if _4bb + 4ph_ in _Ascents_, or _4bb -
4ph_ in _Descents_, be equal to _pp_, there is but one _Elevation_ that
can hit the _Object_, and its _Tangent_ is _pr/2b_. And if _4bb + 4ph_
in _Ascents_, or _4bb - 4ph_ in _descents_, do exceed _pp_, the _Object_
is without the reach of a _Project_ cast with that _Velocity_, and so
the thing impossible.
From this _Equation_ _4bb ∓ 4ph_ = _pp_ are determined the utmost
limits of the reach of any _Project_, and the Figure assigned, wherein
are all the _heights_ upon each _Horizontal distance_ beyond which it
cannot pass; for by reduction of that _Equation_, _h_ will be found =
_¼p_ - _bb/p_ in _heights_, and _bb/p_ - _¼p_ in _descents_; from
whence it follows, that all the Points _h_ are in the _Curve_ of the
_Parabola_, whose _Focus_ is the Point from whence the _Project_ is
cast, and whose _Latus rectum_, or _Parameter ad Axem_ is = _p_.
Likewise from the same _Equation_ may the least _Parameter_ or
_Velocity_ be found capable to reach the _Object_ proposed; for _bb_ =
_¼pp_ ∓ _ph_ being reduced, _½p_ will be = √(_bb + hh_) ± _h_
{in ascents|in descents} which is the _Horizontal Range_ at 45 degrees,
of a Project cast with the least Velocity that would just reach the
_Object_, and the _Elevation_ requisite will be easily had; for dividing
the so found _Semi-parameter_ by the _Horizontal distance_ given _b_,
the _Quote_ into _Radius_ will be the _Tangent_ of the _Elevation_
sought. This Rule may be of good use to all _Bombardiers_ and _Gunners_,
not only that they may use no more Powder than is necessary, to cast
their _Bombs_ into the place assigned, but that they may shoot with much
more certainty, for that a small Error committed in the _Elevation_ of
the _Piece_, will produce no sensible Difference in the fall of the
Shot: For which Reasons the _French_ Engineers in their late Sieges have
used Mortar-pieces inclin'd constantly to the _Elevation_ of 45,
proportioning their Charge of Pouder according to the distance of the
_Object_ they intend to strike on the Horizon.
And this is all that need to be said concerning this _Problem_ of
shooting upon _Heights_ and _Descents_. But if a _Geometrical_
Construction thereof be required; I think I have one that is as easy as
can be expected, which I deduce from the foregoing _Analytical
Solution_, _viz._
_t/r_ = _p/2b_ ± √((_¼pp ± ph - bb_)/_bb_),
and 'tis this, having made the right Angle GDF, (_Tab. 5. Fig. 3._) make
DF = _½p_, or greatest Range, and GD = _b_ the Horizontal Distance,
and DB = _h_ the perpendicular heighth of the Object; to be laid upwards
from D, if the Object be above the Horizon, or downwards if below it.
Parallel to GD draw FA, and make it equal to GB the Hypothenusal
Distance of the Object; and with the Center A and Radius FB = _½p ±
h_, sweep an Arch, which shall if the thing be possible, intersect the
indeterminate Perpendicular DF in two Points K and L, to which draw the
Lines, GL, GK; I say, the Angles DGK, DGL, are the Elevations requisite
to strike the Object B.
_Demonstration._ The Square of FK or FL, is equal to FB_q_ - GB_q_: or
(_½p ± h_)² - _bb_ - _hh_ or _¼pp ± ph - bb_, and therefore
√(_¼pp ± ph - bb_) is = FK = FL, and by Consequence DK, DL =
_½p_ ± √(_½pp ± ph - bb_). And as DG: DK and DL :: Radius:
Tangents sought, which coincides with our Algebraical Expression thereof.
_Prop._ XI. To determine the Force or _Velocity_ of a _Project_, in
every Point of the _Curve_ it describes.
To do this we need no other _Præcognita_, but only the third
Proposition, _viz._ That the _Velocity_ of _falling Bodies_, is double
to that which in the same time, would have described the Space _fallen_
by an equable Motion: For the _Velocity_ of a Project, is compounded of
the constant equal _Velocity_ of the impressed Motion, and the
_Velocity_ of the _Fall_, under a given _Angle_, _viz._ the Complement
of the _Elevation_: For Instance, in _Fig. 2._ in the time wherein a
Project would move from G to L, it descends from L to X, and by the
third _Proposition_ has acquired a _Velocity_, which in that time would
have carried it by an equable Motion from L to Z, or twice the Descent
LX; and drawing the Line GZ, I say, the _Velocity_ in the Point X,
compounded of the _Velocities_ GL and LZ under the Angle GLZ, is to the
_Velocity_ impress'd in the Point G, as GZ is to GL; this follows from
our second _Axiom_, and by the 20 and 21 _Prop. lib. 1. conic.
Midorgii_, XO parallel and equal to GZ shall touch the _Parabola_ in the
Point X. So that the _Velocities_ in the several Points, are as the
lengths of the _Tangents_ to the _Parabola_ in those Points, intercepted
between any two _Diameters_: And these again are as the _Secants_ of the
_Angles_, which those _Tangents_ continued make with the _Horizontal_
Line GB. From what is here laid down, may the comparative Force of a
_Shot_ in any two Points of the _Curve_, be either _Geometrically_ or
_Arithmetically_ discover'd.
_Corollary._
From hence it follows, that the force of a Shot is always least at U, or
the _Vertex_ of the _Parabola_, and that at equal distances therefrom,
as at T and X, G and B its force is always equal, and that the least
force in U is to that in G and B, as _Radius_ to the Secant of the
_Angle_ of _Elevation_ FGB.
These _Propositions_ considered, there is no question relating to
_Projects_, which, by the help of them, may not easily be Solved; and
tho' it be true that most of them are to be met withal, in _Galileus_,
_Torricellius_ and others, who have taken them from those Authors, yet
their Books being Foreign, and not easy to come by, and their
_Demonstrations_ long and difficult, I thought it not amiss to give the
whole _Doctrine_ here in _English_, with such short _Analytical_ Proof
of my own, as might be sufficient to evince their Truth.
The Tenth _Proposition_ contains a _Problem_, untouch'd by _Torricellius_,
which is of the greatest use in _Gunnery_, and for the sake of which
this _Discourse_ was principally intended: It was first Solved by Mr.
_Anderson_, in his Book of the Genuine Use and Effects of the _Gun_,
Printed in the Year 1674; but his Solution required so much Calculation,
that it put me upon search, whether it might not be done more easily,
and thereupon in the Year 1678 I found out the Rule I now Publish, and
from it the _Geometrical_ Construction: Since which time there has a
large _Treatise_ of this Subject, Intituled, _L'Art dejetter les
Bombes_, been Published by _Monsieur Blondel_, wherein he gives the
Solutions of this _Problem_ by _Messieurs Buot_, _Romer_ and _de la
Hire_: But none of them being the same with Mine, or, in my Opinion,
more easy, and most of them more Operose, and besides mine finding the
_Tangent_, which generally determines the _Angle_ better than its
_Sine_, I thought my self obliged to Print it for the use of all such,
as desire to be informed in the _Mathematical_ part of the Art of
_Gunnery_.
Now these Rules were rigidly true, were it not, as I said before, for
the Opposition of the Medium, whereby not only the direct imprest Motion
is continually retarded, but likewise the increase of the _Velocity_ of
the _Fall_, so that the Spaces described thereby, are not exactly as the
Squares of the Times: But what this Opposition of the _Air_ is, against
several _Velocities_, _Bulks_, and _Weights_, is not so easie to
determine. 'Tis certain that the weight of _Air_ to that of _Water_, is
nearly as 1 to 800, whence the weight thereof, to that of any _Project_
is given; 'tis very likely, that to the same _Velocity_ and _Magnitude_,
but of different Matter, the _Opposition_ should be _reciprocally_ as
the weights of the Shot; as likewise that to Shot of the same _Velocity_
and Matter, but of different _Sizes_, it should be as the _Diameters
reciprocally_: Whence generally the _Opposition_ to Shot with the same
_Velocity_, but of differing _Diameters_, and _Materials_, should be as
their _Specifick Gravities_ into their _Diameters reciprocally_; but
whether the _Opposition_, to differing _Velocities_ of the same Shot, be
as the _Squares_ of those _Velocities_, or as the _Velocities_
themselves, or otherwise, is yet a harder Question. However it be, 'tis
certain, that in large Shot of _Metal_, whose weight many Thousand times
surpasses that of the _Air_, and whose force is very great, in
proportion to the _Surface_ wherewith they press thereon; this
_Opposition_ is scarce discernable; For by several _Experiments_ made
with all Care and Circumspection with a _Mortarpiece_, Extraordinary
well fix'd to the Earth on purpose, which carried a solid Brass Shot of
four Inches and a half _Diameter_, and of about fourteen Pound Weight,
the _Ranges_ above and below forty five _Degrees_ were found nearly
equal; if there were any difference, the under _Ranges_ went rather the
farthest, but those differences were usually less than the Errors
committed in ordinary Practice, by the unequal Goodness and Dryness of
the same sort of Powder, by the Unfitness of the Shot to the Bore, and
by the Loosness of the Carriage.
In a smaller Brass-Shot of about an Inch and half Diameter, cast by a
Cross-Bow which ranged it, at most about four Hundred Foot, the Force
being much more equal than in the Mortarpiece, this difference was found
more Curiously: and Constantly and most Evidently, the under Ranges
out-went the upper. From which Trials I conclude, that although in small
and light Shot, the Opposition of the Air, ought and must be accounted
for; yet in Shooting of great and weighty Bombs, there need be very
little or no allowance made; and so these Rules may be put in practice
to all Intents and Purposes, as if this Impediment were absolutely
remov'd.
_A Proposition of general Use in the Art of Gunnery, shewing the Rule of
laying a Mortar to pass, in order to strike an Object above or below the
Horizon._
It was formerly the Opinion of those concerned in Artillery, that there
was a certain requisite of Powder for each Gun, and that in Mortars,
where the distance was to be varied, it must be done by giving a greater
or lesser Elevation to the Piece. But now our later Experience has
taught us that the same thing may be more certainly and readily
performed by increasing and diminishing the quantity of Powder, whether
regard be had to the Execution to be done, or to the Charge of doing it.
For when Bombs are discharged with great Elevations of the Mortar, they
fall too Perpendicular, and bury themselves too deep in the Ground, to
do all that damage they might, if they came more Oblique, and broke upon
or near the Surface of the Earth; which is a thing acknowledg'd by the
Besieged in all Towns, who unpave their Streets, to let the Bombs bury
themselves, and thereby stifle the force of their Splinters. A Second
Convenience is, that at the extream Elevation, the Gunner is not obliged
to be so curious in the direction of his Piece, but it will suffice to
be within a Degree or two of the Truth; whereas in the other Method of
Shooting he ought to be very curious. But a Third, and no less
considerable Advantage is, in the saving the Prince's Powder, which in
so great and so numerous Discharges, as we have lately seen, must needs
amount to a considerable Value. And for Sea-Mortars, it is scarce
practicable otherwise to use them, where the agitation of the Sea
continually changes the Direction of the Mortar, and would render the
Shot very uncertain, were it not that they are placed about 45 Degrees
Elevation, where several Degrees above or under, make very little
difference in the Effect.
In the precedent Discourse, I considered all the Propositions relating
to the Motion of Projectiles, and gave a Solution to this Problem;
_viz._ _To hit an Object above or below the Horizontal Line with the
greatest Certainty and least Force._ That is, that the Horizontal
distance of the Object being put = _b_, and the Perpendicular Heighth =
_b_, the Charge requisite to strike the Object with the greatest
Advantage, was that which with an Elevation of 45° would cast the Shot
on the Horizontal Line, to the distance of √(_bb +hh_), when the
Object was above the Horizon; or if it were below it, the Charge must be
lesser, so as to reach on the Horizon, at 45° Elevation, no greater a
Distance than √(_bb + hh_) - _h_; that is, in the one Case, the Sum
of the Hypothenusal Distance of the Object from the Gun, and the
Perpendicular Heighth thereof above the Gun; and in the other Case, when
the Object is below the Horizon, the difference of the same _per_ 47. I
_Eucl._ And I then shew'd how to find the Elevation proper for the Gun
so charged, _viz._ As the Horizontal Distance of the Object, to the Sum
or Difference of the Hypothenusal Distance and Perpendicular Height: So
Radius to the Tangent of the Elevation sought. But I was not at that
time aware that the aforesaid Elevation did constantly bisect the Angle
between the Perpendicular and the Object, as is demonstrated from the
Difference and Sum of the _Tangent_ and _Secant_ of any Arch being
always equal to the _Tangent_ and _Cotangent_ of the half Complement
thereof to a Quadrant. Having discovered this, I think nothing can be
more compendious, or bids fairer to compleat the Art of Gunnery, it
being as easie to shoot with a Mortar at any Object on demand, as if it
were on the Level; neither is there need of any Computation, but only
simply laying the Gun to pass, in the middle Line between the Zenith and
the Object, and giving it its due Charge. Nor is there any great need of
Instruments for this purpose: For if the Muzzle of the Mortar be turned
truly Square to the Bore of the Piece, as it usually is or ought to be,
a piece of Looking-glass Plate applied parallel to the Muzzle, will by
its Reflection give the true Position of the Piece, the Bombardeer
having no more to do, but to look perpendicularly down on the
Looking-glass, along a small Thread with a Plumbet, and to raise or
depress the Elevation of the Piece, till the Object appear reflected on
the same Point of the _Speculum_, on which the Plumbet falls; for the
Angle of Incidence and Reflection being equal, in this Case a Line at
Right Angles to the _Speculum_, as is the _Axis_ of the Chase of the
Piece, will bisect the Angle between the Perpendicular and the Object,
according as our Proposition requires. So that it only remains by good
and valid Experiments to be assured of the Force of Gunpowder, how to
make and conserve it equal, and to know the Effect thereof in each
Piece; that is, how far differing Charges will cast the same Shot out of
it; which may most conveniently be engraven on the outside thereof, as a
standing Direction to all Gunners, who shall from thence forward have
occasion to use that Piece: And were this Matter well ascertained, it
might be worth the while to make all Mortars of the like Diameter as
near as may he, alike in length of Chase, Weight, Chamber, and all other
Circumstances.
This Discovery that the utmost Range on an inclined Plane, is, when the
_Axis_ of the Piece makes equal Angles with the Perpendicular and the
Object; compared with what I have demonstrated of the same Problem in
the aforesaid Discourse does lead to and discover two very ready
Theorems; the one, to find the greatest Horizontal Range at 45°
Elevation, by any Shot made upon any inclined Plane, with any Elevation
of the Piece whatsoever: And the other to find the Elevations proper to
strike a given Object, with any Force greater than what suffices to
reach it with the aforesaid middle Elevation. Both which being performed
by one single Proportion, may be very serviceable to such as are
concerned in the Practice of Gunnery, but are unwilling to trouble
themselves with tedious and difficult Rules. The two Propositions are
these.
_PROP. I._
A Shot being made on an inclined Plane, having the Horizontal Distance
of the Object it strikes, with the Elevation of the Piece, and the Angle
at the Gun between the Object and the Perpendicular; to find the
greatest Horizontal Range of that Piece, laden with the same Charge;
that is, half the _Latus rectum_ of all the _Parabolæ_ made with the
same _Impetus_.
_RULE._
Take half the Distance of the Object from the _Nadir_, and take the
Difference of the given Elevation from that half; the Versed Sine of
twice that Difference subtract from the Versed Sine of the Distance of
the Object from the _Zenith_: Then shall the Difference of those Versed
Sines be to the Sine of the Distance of the Object from the _Zenith_, as
the Horizontal Distance of the Object strook, to the greatest Horizontal
Range at 45°.
_PROP. II._
Having the greatest Horizontal Range of a Gun, the Horizontal Distance
and Angle of Inclination of an Object to the Perpendicular, to find the
two Elevations necessary to strike that Object.
_RULE._
Halve the Distance of the Object from the _Nadir_; this half is always
equal to the half Sum of the two Elevations we seek. Then say, _As the
greatest Horizontal Range is to the Horizontal Distance of the Object:
So is the Sine of the Angle of Inclination or Distance of the Object
from the Perpendicular, to a fourth Proportional; which fourth being
subtracted from the Versed Sine of the Distance of the Object from the
_Zenith_, leaves the Versed Sine of the Difference of the Elevations
sought; which Elevations are therefore had by adding and subtracting the
half Difference to and from the aforesaid half Sum._
I shall not need to speak of the Facility of these Solutions, I shall
only observe that they are both General, without Exception or Caution,
and derived from the Knowledge that these two Elevations are equidistant
above and below the Line, bisecting the Angle between the Object and the
_Zenith_.
_A Discourse concerning the Measure of the Airs Resistance to Bodies
moved in it. By the Learned _John Wallis_, S. T. D. and R. S. S._
1. That the Air (and the like of any other _Medium_) doth considerably
give Resistance to Bodies moved in it; and doth thereby abate their
Celerity and Force; is generally admitted. And Experience doth attest
it: For otherwise, a Cannon Bullet projected Horizontally, should
(supposing the Celerity and Force undiminished) strike as hard against a
Perpendicular Wall, erected at a great distance, as near at hand; which
we find it doth not.
2. But at what Rate, or in what Proportion, such Resistance is; and
(consequently, at what Rate the Celerity and Force is continually
diminished) seems not to have been so well examined. Whence it is, that
the Motion of a Project (secluding this Consideration) is commonly
reputed to describe a Parabolick Line; as arising from an Uniform or
equal Celerity in the Line of Projection, and a Celerity uniformly
accelerated in the Line of Descent; which two so compounded, do create a
Parabola.
3. In order to the Computation hereof, I first premise this _Lemma_, (as
the most rational that doth occur for my first footing,) That (supposing
other things equal) the Resistance is proportional to the Celerity. For
in a double Celerity, there is to be removed (in the same time) twice as
much Air, (which is a double Impediment) in a treble, thrice as much;
and so in other Proportions.
4. Suppose we then the Force impressed (and consequently the Celerity,
if there were no Resistance) as 1; the Resistance as _r_. (which must be
less than the Force, or else the Force would not prevail over the
Impediment, to create a Motion.) And therefore the effective Force at a
first Moment, is to be reputed as 1 - _r_: That is, so much as the Force
impressed, is more than the Impediment or Resistance.
5. Be it as 1 - _r_ to 1; so one to _m_. (which _m_ is therefore greater
than 1.)
6. And therefore the effective Force (and consequently the Celerity) as
to a first Moment, is to be 1/_m_ of what it would be, had there been no
Resistance.
7. This 1/_m_ is also the remaining Force after such first Moment; and
this remaining Force is (for the same Reason) to be proportionally
abated as to a second Moment; that is, we are to take 1/_m_ thereof,
that is 1/_mm_ of the impressed Force. And for a third Moment (at equal
distance of time) 1/_mmm_; for a fourth 1/_m_⁴; and so onward
infinitely.
8. Because the length dispatched (in equal times) is proportional to the
Celerities; the Lines of Motion (answering to those equal Times) are to
be as 1/_m_, 1/_m_², 1/_m_³, 1/_m_⁴, _&c._ of what they would have
been, in the same Times, had there been no Resistance.
9. This therefore is a Geometrical Progression; and (because of _m_
greater than 1) continually decreasing.
10. This decreasing Progression infinitely continued (determining in the
same Point of Rest, where the Motion is supposed to expire) is yet of a
finite Magnitude; and equal to 1/(_m_ - 1), of what it would have been
in so much Time, if there had been no Resistance. As is demonstrated in
my Algebra, _Chap._ 95. _Prop._ 8. For (as I have elsewhere
demonstrated) the Sum or Aggregate of a Geometrical Progression is (_VR
- A_)/(_R_ - 1) (supposing _V_ the greatest Term, _A_ the least, and _R_
the common Multiplier.) That is _VR_/(_R_ - 1) - _A_/(_R_ - 1). Now in
the present Case, (supposing the Progression infinitely continued) the
least Term _A_, becomes infinitely small, or = 0. And consequently
_A_/(_R_ - 1) doth also vanish, and thereby the Aggregate becomes =
_VR_/(_R_ - 1). That is (as will appear by dividing _VR_ by _R_ - 1;) _V
+ V/R + V/RR + V/R³ + &c._ = _VR_/(_R_ - 1);[14] (supposing the
Progression to begin at _V_ = 1.) That is (dividing all by _R_, that so
the Progression may begin at _V/R_ = 1/_m_:) _V_/(_R_ - 1) = _V/R + V/RR
+ V/R³ + &c._, That is, in our present Case (because of _V_ = 1, &
_R_ = _m_:) 1/_m_ + 1/_mm_ + 1/_m_³ &c. = 1/(_m_ - 1). That is,
(putting _n_ = _m_ - 1) 1/_n_ of what it would have been if there had
been no Resistance.
11. This infinite Progression is fitly expressed by an Ordinate in the
Exterior Hyperbola, parallel to one of the Asymptotes; and the several
Members of that, by the several Members of this, cut in continual
Proportion. As is there demonstrated at _Prop._ 15. For let _SH_,
(_vid._ Fig. 4. Tab. 5.) be an Hyperbola between the Asymptotes _AB_,
_AF_: And let the Ordinate _DH_ (in the Exterior Hyperbola, parallel to
_AF_,) represent the impressed Force undiminished; or the Line to be
described in such time, by a Celerity answerable to such undiminished
Force. And let _BS_ (a like Ordinate) be 1/_m_ thereof; which therefore,
being less than _DH_ (as being equal to a Part of it) will be farther
than it from _AF_. In _AB_ (which I put = 1) let _Bd_ be such a Part
thereof, as is _BS_ of _DH_. Now because (as is, well known) all the
inscribed Parallelograms, in the Exterior Hyperbola, _AS_, _AH_, _&c._
are equal; and therefore their sides reciprocal: Therefore as _Ad_ = 1 -
1/_m_ (supposing _Bd_ to be taken, from _B_ towards _A_,) to _AB_ = 1,
or as _m_ - 1 to _m_: so is _BS_ = (1/_m_)_DH_, to _dh_, which is
therefore equal to 1/(_m_ - 1) of _DH_; that is (as will appear by
dividing 1, by _m_ - 1,) to 1/_m_ + 1/_mm_ + 1/_m_³, _&c._ of _DH_.[15]
Or if _Bd_ be taken beyond _B_; then as _Ad_ = 1 + 1/_m_ to _AB_ = 1, or
as _m_ + 1 to _m_, so is (1/_m_)_DH_ to _dh_, which is therefore equal
to 1/(_m_ + 1)DH; that is (as will appear by like dividing of 1 by _m_ +
1;) = to 1/_m_ - 1/_mm_ + 1/_m_³ - _&c._ of _DH_.
12. Let such ordinate _dh_, or (equal to it in the Asymptote) _AF_, be
so divided in _L_, _M_, _N_, _&c._ (by Perpendiculars cutting the
Hyperbola in _l_, _m_, _n_, _&c._) as that _FL_, _LM_, _MN_, be as
1/_m_, 1/_mm_, 1/_m_³, _&c._ That is, so continually decreasing as that
each Antecedent be to its Consequent, as 1 to 1/_m_, or as _m_ to 1. See
_Fig. 5. Tab. 5._
13. This is done by taking _AF_, _AL_, _AN_, _&c._ in such proportion.
For, of continual Proportionals, the Differences are also continually
proportional, and in the same proportion. For let _A_, _B_, _C_, _D_,
_&c._ be such Proportionals, and their Differences _a_, _b_, _c_, _&c._
That is, _A_ - _B_ = _a_, _B_ - _C_ = _b_, _C_ - _D_ = _c_, _&c._
Then, because A, B, C, D, _&c._ are in continual proportion,
That is, A. B :: B. C :: C. D :: _&c._
And dividing (A - B). B :: (B - C). C :: (C - D). D :: _&c._
That is, _a_. B :: _b_. C :: _d_. D :: _&c._
And alternly _a. b. c._ _&c._ :: B. C. D. _&c._ :: A. B. C. _&c._
That is, in continual proportion as A to B, or as _m_ to 1.
14. This being done; the Hyperbolick Spaces _Fl_, _Lm_, _Mn_, &c. are
equal. As is demonstrated by _Gregory San-Vincent_; and as such is
commonly admitted.
15. So that _Fl_, _Lm_, _Mn_, _&c._ may fitly represent equal Times, in
which are dispatched unequal Lengths, represented by _FL_, _LM_, _MN_,
_&c._
16. And because they are in Number infinite (though equal to a finite
Magnitude) the Duration is infinite: And consequently the impressed
Force, and Motion thence arising, never to be wholly extinguished
(without some further Impediment) but perpetually approaching to _A_, in
the Nature of Asymptotes.
17. The Spaces _Fl_, _Fm_, _Fn_, &c. are therefore as Logarithms (in
Arithmetical Progression increasing) answering to the Lines _AF_, _AL_,
_AM_, &c. or to _FL_, _LM_, _MN_, &c. in Geometrical Progression
decreasing.
18. Because _FL_, _LM_, _MN_, &c. are as 1/_m_, 1/_mm_, 1/_m_³, &c.
(infinitely) terminated at _A_; therefore (by ¶ 10) their Aggregate _FA_
or _dh_, is to _DH_, (so much Length as would have been dispatched, in
the same time, by such impressed Force undiminished) as 1 to _m_ - 1 =
_n_.
19. If therefore we take, as 1 to _n_, so _AF_ to _DH_; this will
represent the Length to be dispatched, in the same time, by such
undiminished Force.
20. And if such _DH_ be supposed to be divided into equal Parts
innumerable (and therefore infinitely small;) these answer to those (as
many) Parts unequal in _FA_, or _hd_.
21. But, what is the Proportion of _r_ to 1, or (which depends on it) of
1 - _r_ to 1, or 1 to _m_; remains to be inquired by Experiment?
22. If the Progression be not infinitely continued; but end (suppose) at
_N_, and its least Term be _A_ = _MN_; then, out of
_V_/(_R_ - 1) = 1/_m_ + 1/_mm_ + 1/_m_³, _&c._
is to be subducted _A_/(_R_ - 1) (as at ¶ 10.) that is (as by Division
will appear)
_A_/_R_ + _A_/_R_² + _A_/_R_³ &c.