SECTION II.[22]

But let us now consider the properties of the demonstrative syllogism,

and endeavour to unravel its intricate web; appointing Aristotle for

our guide in this arduous investigation. For an enquiry of this kind

is naturally connected with the doctrine of ideas, as it enables us to

gain a glimpse of the universals participated in mathematical forms,

and to rise to the principles of science. It brings us acquainted with

the laws which bind demonstration; and teaches us that objects of

intellect are alone the objects of science, and the sources of truth.

Previous to the acquisition of all learning and ratiocinative

discipline, it is necessary we should possess certain natural

principles of knowledge, as subservient to our future progress and

attainments. Thus, in every science there are some things which require

an immediate assent as soon as proposed; whose certainty is too evident

and illustrious to stand in need of any demonstrative proof deduced

from that particular science which, like stately pillars, they equally

support and adorn. Hence we are informed by the geometrician, that a

point is that which is destitute of all parts whatever; but we must

previously understand the meaning of the word _part_. Thus the

arithmetician defines an odd number, that which is divided according

to unequal parts; but it is necessary we should antecedently know

the meaning of the word _unequal_. Thus, too, art as well as

science operates by antecedent knowledge; and hence the architect,

the statuary, and the shipwright, learn the names and the use of their

respective implements, previous to the exercise of the materials

themselves. This is particularly evident in the discursive arts of

rhetoric and logic; thus the logician reasons by syllogism, the

rhetorician by induction, and the sophist by digressions and examples;

while each proceeds in an orderly progression from principles simple

and evident, to the most remote and complicated conclusions.

2. The antecedent knowledge of things may be divided into two parts:

the one a knowledge of their existence, or that they exist; the other

a knowledge of the terms expressive of their existence. Thus, previous

to the enquiry why iron is attracted by the magnet, it is necessary

we should learn the reality of this attraction, and the general mode

of its operation: thus too, in an enquiry concerning the nature of

motion and time, we must be previously convinced of their existence

in the nature of things. The second division of antecedent knowledge

takes place in subjects whose very existence admits of a dispute: thus

previous to a solution of the questions, Whether there are any gods

or not? Whether there is a providence or not? and the like, it is

necessary we should first understand the meaning of the terms; since

we in vain investigate the nature of any thing while we are ignorant

of the meaning of its name; although, on the contrary, we may have a

perfect conception of the meaning of some words, and yet be totally

ignorant whether the things they express have a real, or only an

imaginary existence. Thus, the meaning of the word _centaur_ is

well understood by every one; but its existence is questioned by most.

3. From hence it will easily appear, that no small difference subsists

between _learning_ and _knowledge_. He who is about to understand the

truth of any proposition, may be said to possess a previous conception

of its truth; while, on the contrary, it may happen that he who is in

the capacity of a learner, has no antecedent knowledge of the science

he is about to learn. Thus we attain to the distinct knowledge of a

thing which we formerly knew in a general way; and frequently, things

of which we were ignorant are learned and known in the same instant.

Of this kind are the things contained under some general idea, of which

we possess a previous knowledge: thus, he who already knows that the

three interior angles of every triangle are equal to two right, and

is as yet ignorant that some particular figure delineated on paper

is a triangle, is no sooner convinced from inspection of its being

a triangle, than he immediately learns and knows: he learns it is a

triangle; he knows the equality of its angles to two right ones. That

it is now a triangle he both sees and learns; but the equality of its

angles he previously knew in that general and comprehensive idea, which

embraces every particular triangle.

Indeed, a definite knowledge of this triangle requires two conditions:

the one, that it is a triangle; and the other, that it has angles

equal to two right. The first we receive from inspection; the second

is the result of a syllogistic process; an operation too refined for

the energies of sense, and alone the province of _intellect_

and _demonstration_. But demonstration without the knowledge of

that which is universal, cannot subsist; and since the proposition

is universal, that in every triangle the angles are equal to two

right; as soon as any figure is acknowledged to be a triangle, it must

necessarily possess this general property.

Hence we infer, that of the triangle delineated on paper, and

concealed, we are partly ignorant of this general property, the

equality of its angles (because we are ignorant of its existence);

and we partly understand it as included in that universal idea we

previously possessed. Hence too, it is evident that actual science

arises from a medium between absolute ignorance and perfect knowledge;

and that he who possesses the principles of demonstration, possesses in

capacity the conclusions also, however complicated and remote; and that

by an evocation of these principles from dormant power into energy, we

advance from general and abstracted knowledge to that which is sensible

and particular.

4. Two acceptations of knowledge may be admitted; the one common and

without any restriction; the other limited and peculiar. Since all

knowledge, whether arising from accidents, or supported by necessary

principles, is called science. Knowledge, properly so called, arises

from a possession of that cause from which a thing derives its

existence, and by which we infer the necessity of its existence; and

this constitutes simple and absolute science. Thus too, the definitions

of those general conceptions and suppositions, which from their

primary nature are incapable of demonstration, are called science.

But the science which treats of the method of arriving at knowledge,

is called demonstration; for every demonstration is a syllogism

producing science. Hence, if in every syllogism it is necessary that

the propositions should be the cause of the conclusion; and to know

any thing properly, a knowledge of its cause is requisite; in the

propositions of demonstration, both these conditions are required: that

they should be effective of the conclusion; and the causes of the thing

demonstrated.

Thus, from the ruins of a stately edifice, we may justly infer, that

the building was beautiful when entire; and from the smoke we may

collect the existence of the fire, though concealed: but the ruins of

the edifice are not the cause of its beauty; nor does fire originate

from smoke, but, on the contrary, smoke is the natural result of fire:

the inference, therefore, is in neither case a demonstrative one.

Again, since every cause is both prior to, and more excellent than its

effect, it is necessary that the propositions should be more peculiar,

primary, and excellent than the conclusions. And because we then know

a thing properly when we believe it to have a necessary existence,

hence it is requisite that the propositions should be true; for if

false, a false conclusion may ensue, such as, that the diameter of a

square is commensurable with its side. But if every science arises

from antecedent knowledge, demonstration must be founded on something

previous; and on this account it is requisite that the propositions

should be more known than the conclusions. The necessary properties,

then, of all demonstrative propositions, are these; that they exist

as causes, are primary, more excellent, peculiar, true, and known,

than the conclusions. Indeed, every demonstration not only consists of

principles prior to others, but of such as are eminently first; for

if the assumed propositions may be demonstrated by other assumptions,

such propositions may, indeed, appear prior to the conclusions, but

are by no means entitled to the appellation of first. But others, on

the contrary, which require no demonstration, but are of themselves

probable or manifest, are deservedly esteemed the first, the truest,

and the best. Such indemonstrable truths were called by the ancients,

axioms, from their majesty and authority; as the assumptions which

constitute the best syllogisms derive all their force and efficacy from

these.

And on this account, above all others, they merit the title of

the principles of demonstration. But here it is worth observing,

that these primary propositions are not the first in the order of

our conceptions; but first to nature, or in the nature of things.

To us, that which is first is particular, and subject to sensible

inspection; to nature, that which is universal, and far remote from

the apprehension of sense. Demonstration does not submit itself to the

measure of our ingenuity, but, with invariable rectitude, tends to

truth as its ultimate aim; and without stopping to consider what our

limited powers can attain, it alone explores and traces out the nature

of a thing, though to us unperceived and unknown.

This demonstrative syllogism differs not a little from others, by the

above property; the rest can as well educe a true conclusion from false

premises, which is frequent among the rhetoricians, as that which is

prior from that which is posterior; such as, Is every syllogism derived

from conjecture?

With respect to the rest, as we have already confessed, they may be

formed from principles that are true, but not from such as are proper

and peculiar; as if a physician should endeavour to prove an orbicular

wound the most difficult to coalesce and heal, because its figure is of

all others, the most capacious; since the demonstration of this is not

the province of the physician, but of the geometrician _alone_.

5. That proposition is called immediate, which has none superior to

itself, and which no demonstration whatever can confirm: such as these

are held together by the embraces of universals. There are some,

indeed, united from that which is sensible and particular: thus, that

the garment is white, is an immediate proposition, but not of that

kind whose principles require to be demonstrative ones; the cause

of which we shall hereafter investigate. Of immediate propositions

subservient to the purposes of demonstration, some are of such a

superior nature, that all men possess a knowledge of them without any

previous instruction; and these are called axioms, or general notions;

for without these all knowledge and enquiry is vain. Another species

of immediate propositions is position; incapable of being strengthened

by demonstration, yet not necessarily foreknown by the learner, but

received from the teacher. With respect to the genus of position, one

of its species is definition, and another hypothesis. Definition is an

oration, in which we neither speak of the existence, nor non-existence

of a thing; but alone determine its nature and essence. It is common to

every hypothesis, not to be derived from nature, but to be the entire

result of the art of the preceptor.

It likewise always affirms the existence or non-existence of its

subject: such as, that motion is, and that from nothing nothing is

produced. Those which are not so perspicuous are called postulates, or

petitions; as that a circle may be described from any centre, and with

any radius; and such as these are properly hypotheses and postulates.

6. We have now seen the privilege assigned to the principles of

demonstration:--whether or no our decision has been just, the

ensuing considerations will evince. We said that the assumptions in

demonstration were more known than the conclusions,--not indeed without

reason, since through these our knowledge and belief of the conclusion

arises. For universally, that quality which is attributed to many

different things so as to be assigned to one through the medium of

another, abounds most in that medium by which it is transmitted to the

rest.

Thus the sun, through the medium of the moon, illuminates the earth by

night; thus the father loves the preceptor through the medium of his

child. And in the first instance the moon is more lucid than any object

it enlightens: in the second, the child possesses more of the father’s

regard than his preceptor. If then we assent to the conclusions through

our belief of the principles alone, it is necessary that the principles

should be more known, and inherit a greater degree of our assent.

Hence, if it be true that the principles are more known than the

conclusions, it follows, that either our knowledge of them is derived

from demonstration, or that it is superior to any demonstrative proof;

and after this manner we must conceive of those general self-evident

notions which, on account of their indemonstrable certainty, are

deservedly placed at the top of all human science.

These propositions not only possess greater credibility than their

conclusions, they likewise inherit this property as an accession to

their dignity and importance; that no contrary propositions deserve

greater belief; for if you give no more assent to any principle than

to its contrary, neither can you give more credit to the conclusion

deduced from that principle than to its opposite. Were this the

case, the doctrine of these propositions would immediately lose its

invariable certainty.

7. There are, indeed, some who, from erroneously applying what we

have rightly determined, endeavour to take away the possibility

of demonstration. From the preceding doctrine it appears that the

principles are more aptly known than the conclusions. This is not

evident to some, who think nothing can be known by us without a

demonstrative process; and consequently believe that the most

simple principles must derive all their credit from the light of

demonstration.

But if it be necessary that all assumptions should be demonstrated by

others, and these again by others; either the enquiry must be continued

to infinity, (but infinity can never be absolved), or if, wearied by

the immense process, you at length stop, you must doubtless leave those

propositions unknown, whose demonstration was declined through the

fatigue of investigation. But how can science be derived from unknown

principles? For he who is ignorant of the principles, cannot understand

the conclusions which flow from these as their proper source, unless

from an hypothesis or supposition of their reality.

This argument of the sophists is, indeed, so far true, that he who does

not understand that which is first in the order of demonstration, must

remain ignorant of that which is last:--But in this it fails, that

all knowledge is demonstrative; since this is an assertion no less

ridiculous than to maintain that nothing can be known. For as it is

manifest that some things derive their credit and support from others,

it is equally obvious that many, by their intrinsic excellence, possess

indubitable certainty and truth; and command our immediate assent as

soon as proposed. They inherit, indeed, a higher degree of evidence

than those we assent to by the confirmation of others; and these are

the first principles of demonstration: propositions indisputable,

immediate, and perspicuous by that native lustre they always possess.

By means of these, we advance from proposition to proposition, and

from syllogism to syllogism, till we arrive at the most complicated

and important conclusions. Others, willing to decline this infinite

progression, defend the necessity of a circular or reciprocal

demonstration. But this is nothing more than to build error upon error,

in order to attain the truth; an attempt no less ridiculous than that

of the giants of old. For since, as we shall hereafter accurately

prove, demonstration ought to consist from that which is first, and

most known; and since it is impossible that the same thing should be

to itself both prior and posterior: hence we infer the absurdity of

circular demonstration; or those syllogisms in which the conclusions

are alternately substituted as principles, and the principles as

conclusions. It may, indeed, happen, that the same thing may be both

prior and posterior to the same; but not at one and the same time, nor

according to the same mode of existence. Thus, what is prior in the

order of our conceptions, is posterior in the order of nature; and what

is first in the arrangement of things, is last in the progressions of

human understanding. But demonstration always desires that _first_

which is prior in the order and constitution of nature. But the folly

of such a method will more plainly appear from considering its result:

let us suppose every _a_ is _b_, and every _b_ is _c_; hence we justly

infer, that every _a_ is _c_. In like manner, if we prove that every

_a_ is _b_, and by a circular demonstration, that every _b_ is _a_, the

consequence from the preceding is no other than that every _a_ is _a_;

and thus the conclusion terminates in that from which it first began; a

deduction equally useless and ridiculous. However, admitting that, in

the first figure, circular demonstration may be in some cases adopted,

yet this can but seldom happen from the paucity of reciprocal terms.

But that reciprocal terms are very few, is plain from hence: let

any species be assumed, as _man_; whatever is the predicate of

man, is either constitutive of his essence, or expressive of some

accident belonging to his nature. The superior genera and differences

compose his essence, among which no equal predicate can be assigned

reciprocable with man, except the ultimate differences which cannot be

otherwise than one, i. e., risibility, which mutually reciprocates

with its subject; since every man is risible, and whatever is risible

is man. Of accidents some are common, others peculiar; and the common

are far more in number than the peculiar; consequently the predicates

which reciprocate with man, are much fewer than those which do not

reciprocate.

8. It is now necessary to enumerate the questions pertaining to

demonstration; and for this purpose, we shall begin with propositions,

since from these, syllogisms are formed; and since every proposition

consists of a subject and predicate, the modes of predication must be

considered, and these are three which I call _total_, _essential_,

and _universal_; a _total_ predication takes place when that which is

affirmed or denied of one individual is affirmed or denied of every

individual comprehended under the same common species.

Thus, animal is predicated of every man, and it has this farther

property besides, that of whatever subject it is true to affirm man, it

is at the same time true to affirm animal.

Those things are said to be _essentially_ predicated; first,

when the predicate is not only total, but constitutes the essence

of the subject; instances of this kind are, animal of man; tree of

the plantain; a line of a triangle; for a triangle is that which is

contained under three right-lines. But here we must observe, that

not every total predicate is an essential one; thus, whiteness is

predicated of every swan, because it is inherent in every swan, and at

every instant of time; but because whiteness does not constitute the

essence of a swan, it is not essentially predicated; and this, first,

is one of the modes of essential predication of the greatest importance

in demonstration. The second mode is of accidents, in the definition

of which their common subject is applied: thus, a line is essentially

inherent in rectitude, because in its geometrical definition, a line

is adopted; for rectitude is no other than a measure, equally extended

between the points of a line. In the same manner, imparity is contained

in number; for what is that which is odd, but a number divided into

unequal parts? Thus, virtues are resident in the soul, because, in

their definition, either some part of the soul, or some one of its

powers, is always applied. The third mode of essential predicates

pertains to accidents which are inseparably contained in some

particular subject, so as to exclude a prior existence in any other

subject; such as colour in superficies. The fourth mode is of things

neither contained in another, nor predicated of others; and such are

all individuals, as Callias, Socrates, Plato. Causes are likewise said

to exist substantially, which operate neither from accident nor

fortune.

Thus, digging up the ground for the purposes of agriculture, may be the

cause of discovering a treasure, but it is only an accidental one. But

the death of Socrates, in despite of vigilance, is not the result of a

fortuitous cause, but of an essential one, viz. the operation of poison.

9. These posterior significations of essential predicates are added

more for the sake of ornament than use; but the two former have a

necessary existence, since they cannot but exist in the definition of

names which predicate the essence of a thing, and in subjects which are

so entirely the support of accidents, that they are always applied in

their definition. But it is a doubt with some, whether those accidents

are necessary, which cannot be defined independent of their common

subject? To this we answer, that no such accident can, from its nature,

be contained in every individual of any species; for curvature is not

contained in every line; nor imparity in every number; from whence we

infer, that neither is curvature necessarily existent in a line, nor

parity in number. The truth of this is evident from considering these

accidents abstracted from their subjects; for then we shall perceive

that a line may exist without curvature, and number without imparity.

Again, I call that an _universal_ predicate, which is predicated

of a subject totally and essentially, and considered as primarily

and inseparably inherent in that subject: for it does not follow

that a predicate, which is total, should be immediately universal;

for whiteness is affirmed of every swan, and blackness of every

crow, yet neither universally. In like manner, a substantial

predicate is not consequently an universal one; for the third mode

of essential predicates, and the two following (instanced before)

cannot be universal. Thus, colour, although inherent in superficies

essentially, is not inherent in every superficies, and consequently

not universally. Thus again, Socrates, Callias, and Plato, though they

exist essentially, are not universals, but particulars; and thus,

lastly, the drinking of poison was an essential cause of the death of

Socrates, but not an universal one, because Socrates might have died

by other means than poison. If then, we wish to render an accurate

definition of an universal predicate, we must not only say it is total

and essential, but that it is primarily present to its subject and no

other. Thus, the possession of angles equal to two right, primarily

belongs to a triangle; for this assertion is essentially predicated of

triangle, and is inherent in every triangle. This property, therefore,

is not universally in figure, because it is not the property of every

figure, not of a square, for instance; nor as universal in a scalene

triangle: for although it is contained in every scalene, and in every

equilateral, and isosceles triangle, yet it is not primarily contained

in them, but in triangle itself; because these several figures inherit

this property, not from the particular species to which they belong,

but from the common genus _triangle_. And thus much concerning

_total_, _essential_, and _universal_ predicates.

10. Concerning that which is universal, we are frequently liable to

err; often from a belief that our demonstration is universal, when it

is only particular; and frequently from supposing it particular when it

is, on the contrary, universal. There are three causes of this mistake;

the first, when we demonstrate any particular property of that which is

singular and individual, as the sun, the earth, or the world. For since

there is but one sun, one earth, and one world, when we demonstrate

that the orb of the earth possesses the middle place, or that the

heavens revolve, we do not then appear to demonstrate that which is

universal.

To this we answer: when we demonstrate an eclipse of the sun to arise

from the opposition of the moon, we do not consider the sun as one

particular luminary, but we deduce this consequence as if many other

suns existed besides the present.

Just as if there were but one species of triangles existed; for

instance, the _isosceles_; the equality of its angles at the base

would not be considered in the demonstration of the equality of all its

angles to two right ones: but its triangularity would be essential,

supposing every species of triangles but the isosceles extinct, and no

other the subject of this affection. So when we prove that the sun is

greater than the earth, our proof does not arise from considering it

as this particular sun alone, but as sun in general; and by applying

our reasoning to every sun, if thousands besides the present should

enlighten the world. This will appear still more evident, if we

consider that such conclusions must be universal, as they are the

result of an induction of particulars: thus, he who demonstrates that

an eclipse of the sun arises from the opposition of the moon between

the sun and earth, must previously collect, by induction, that when any

luminous body is placed in a right-line with any two others opaque,

the lucid body shall be prevented, in a greater or less degree, from

enlightening the last of these bodies, by the intervention of the

second; and by extending this reasoning to the sun and earth, the

syllogism will run thus:

Every lucid body placed in a right-line with two others opaque,

will be eclipsed in respect of the last by the intervention of

the second;

The sun, or every sun, is a luminous body with these conditions;

And consequently the sun, and so every sun, will be eclipsed to

the earth by the opposition of the moon.

Hence, in cases of this kind, we must ever remember, that we

demonstrate no property of them as singulars, but as that universal

conceived by the abstraction of the mind.

Another cause of deception arises, when many different species agree in

one ratio or analogy, yet that in which they agree is nameless. Thus

number, magnitude, and time, differ by the diversities of species;

but agree in this, that as any four comparable numbers correspond

in their proportions to each other, so that as the first is to the

second, so is the third to the fourth; or alternately, as the first

to the third; so is the second to the fourth: in a similar manner,

four magnitudes, or four times, accord in their mutual analogies and

proportions. Hence, alternate proportion may be attributed to lines

as they are lines, to numbers as they are numbers, and afterwards to

_times_ and to bodies, as the demonstration of these is usually

separate and singular; when the same property might be proved of all

these by one comprehensive demonstration, if the common name of their

genus could be obtained: but since this is wanting, and the species

are different, we are obliged to consider them separately and apart;

and as we are now speaking of that universal demonstration which is

properly one, as arising from one first subject; hence none of these

obtain an universal demonstration, because this affection of alternate

proportion is not restricted to numbers or lines, considered in

themselves, but to that common something which is supposed to embrace

all these, and is destitute of a proper name. Thus too we may happen

to be deceived, should we attempt to prove the equality of three

angles to two right, separately, of a scalene, an isosceles, and an

equilateral triangle, only with this difference, that in the latter

case the deception is not so easy as in the former; since here the

name triangle, expressive of their common genus, is assigned. A third

cause of error arises from believing that to demonstrate any property

inherent after some particular manner in the whole of a thing, is

to demonstrate that property universally inherent. Thus, geometry

proves[23] that if a right-line falling upon two right-lines makes

the outward angle with the one line a right-angle, and the inward

and opposite angle with the other a right one, those two right-lines

shall be parallel, or never meet, though infinitely extended. This

property agrees to all lines which make right-angles: but they are not

primarily equidistant on this account, since, if they do not each make

a right-angle, but the two conjointly are equal to two right, they

may still be proved equidistant. This latter demonstration, then, is

primarily and universally conceived; the other, which always supposes

the opposite angles right ones, does not conclude universally; though

it concludes totally of all lines with such conditions: the one may

be said to conclude of a greater _all_; the other of a lesser.

It is this greater _all_ which the mind embraces when it assents

to any self-evident truth; or to any of the propositions of Euclid.

But by what method may we discover whether our demonstration is of

this greater or lesser all? We answer, that general affection which

constitutes universal demonstration is always present to that subject,

which when taken away, the predicate is immediately destroyed, because

the first of all its inherent properties.

Thus, for instance, some particular sensible triangle possesses these

properties:--it consists of brass; it is scalene; it is a triangle. The

query is, by which of these we have just now enumerated, this affection

of possessing angles equal to two right is predicated of the triangle?

Take away the brass, do you by this means destroy the equality of its

angles to two right ones? Certainly not:--take away its scalenity, yet

this general affection remains: lastly, take away its triangularity,

and then you necessarily destroy the predicate; for no longer can this

property remain, if it ceases to be a triangle.

But perhaps some may object from this reasoning, such a general

affection extends to figure, superficies, and extremities, since, if

any of these are taken away, the equality of its angles to two right

can no longer remain. It is true, indeed, that by a separation of

figure, superficies, and terms, from a body, you destroy all the modes

and circumstances of its being; yet not because these are taken away,

but because the triangle, by the separation of these, is necessarily

destroyed; for if the triangle could still be preserved without figure,

superficies, and terms, though these were taken away it would still

retain angles equal to two right; but this is impossible. And if all

these remain, and triangle is taken away, this affection no longer

remains. Hence the possession of this equality of three angles to two

right, is primarily and universally inherent in triangle, since it is

not abolished by the abolition of the rest:----such as to consist of

brass; to be scalene, or the like. Neither does it derive its being

from the existence of the rest alone; as figure, superficies, terms;

since it is not every figure which possesses this property, as is

evident in such as are quadrangular, or multangular. And thus it is

preserved by the preservation of triangle, it is destroyed by its

destruction.

11. From the principles already established, it is plain that

demonstration must consist of such propositions as are universal and

necessary. That they must be universal, is evident from the preceding;

and that they must be necessary, we gather probably from hence; that in

the subversion of any demonstration we use no other arguments than the

want of necessary existence in the principles.

We collect their necessity demonstratively, thus; he who does not

know a thing by the proper cause of its existence, cannot possess

science of that thing; but he who collects a necessary conclusion from

a medium not necessary, does not know it by the proper cause of its

existence, and therefore he has no proper science concerning it. Thus,

if the necessary conclusion _c_ is _a_, be demonstrated by

the medium B, not necessary; such a medium is not the cause of the

conclusion; for since the medium does not exist necessarily, it may

be supposed not to exist; and at the time when it no longer exists,

the conclusion remains in full force; because, since necessary,

it is eternal. But an effect cannot exist without a cause of its

existence; and hence such a medium can never be the cause of such a

conclusion. Again, since in all science there are three things, with

whose preservation the duration of knowledge is connected; and these

are, first, he who possesses science; secondly, the thing known; and

thirdly, the reason by which it is known; while these endure, science

can never be blotted from the mind, but on the contrary, if science be

ever lost, it is necessary some of these three must be destroyed.

If then you infer that the science of a necessary conclusion may be

obtained from a medium not necessary, suppose this medium, since

capable of extinction, to be destroyed; then the conclusion, since

necessary, shall remain; but will be no longer the object of knowledge,

since it is supposed to be known by that medium which is now extinct.

Hence, science is lost, though none of the preceding three are

taken away; but this is absurd, and contrary to the principles we

have just established. The thing known remains; for the conclusion,

since necessary, cannot be destroyed;--he who knows still remains,

since neither dead, nor forgetful of the conclusion:--lastly, the

demonstration by which it was known, still survives in the mind; and

hence we collect, that if science be no more after the corruption

of the medium, neither was it science by that medium before its

corruption; for if science was ever obtained through such a medium,

it could not be lost while these three are preserved. The science,

therefore, of a necessary conclusion can never be obtained by a medium

which is not necessary.

12. From hence it is manifest, that demonstrations cannot emigrate from

one genus to another; or by such a translation be compared with one

another. Such as, for instance, the demonstrations of geometry with

those of arithmetic. To be convinced of this, we must rise a little

higher in our speculations, and attentively consider the properties

of demonstration: one of which is, that predicate which is always

found in the conclusion, and which affirms or denies the existence of

its subject: another is, those axioms or first principles by whose

universal embrace demonstration is fortified; and from whose original

light it derives all its lustre. The third is the subject genus, and

that nature of which the affections and essential properties are

predicated; such as magnitude and number. In these subjects we must

examine when, and in what manner a transition in demonstrations from

genus to genus may be allowed. First, it is evident, that when the

genera are altogether separate and discordant, as in arithmetic and

geometry, then the demonstrations of the one cannot be referred to

the other. Thus, it is impossible that arithmetical proofs can ever

be accommodated with propriety to the accidents of magnitudes: but

when the genera, as it were, communicate, and the one is contained

under the other, then the one may transfer the principles of the other

to its own convenience. Thus, optics unites in amicable compact with

geometry, which defines all its suppositions; such as lines that are

_right_, angles acute, lines equilateral, and the like. The same

order may be perceived between arithmetic and music: thus, the double,

sesquialter, and the like, are transferred from arithmetic, from which

they take their rise, and are applied to the measures of harmony.

Thus, medicine frequently derives its proofs from nature, because

the human body, with which it is conversant, is comprehended under

natural body. From hence it follows, that the geometrician cannot, by

any geometrical reasons demonstrate any truth, abstracted from lines,

superficies, and solids; such as, that of contraries there is the same

science; or that contraries follow each other; nor yet such as have an

existence in lines and superficies, but not an essential one, in the

sense previously explained.

Of this kind is the question, whether a right-line is the most

beautiful of lines? or whether it is more opposed to a line perfectly

orbicular, or to an arch only. For the consideration of beauty, and the

opposition of contraries, does not belong to geometry, but is alone the

province of metaphysics, or the first philosophy.

But a question here occurs, If it be requisite that the propositions

which constitute demonstration should be peculiar to the science they

establish, after what manner are we to admit in demonstration those

axioms which are conceived in the most common and general terms; such

as, if from equal things you take away equals, the remainders shall be

equal:----as likewise, of every thing that exists, either affirmation

or negation is true? The solution is this: such principles, though

common, yet when applied to any particular science for the purposes

of demonstration, must be used with a certain limitation. Thus the

geometrician applies that general principle, if from equal things, &c.

not simply, but with a restriction to magnitudes; and the arithmetician

universally to numbers.

Thus too, that other general proposition:----of every thing,

affirmation or negation is true; is subservient to every art, but not

without accommodation to the particular science it is used by. Thus

number _is_ or is _not_, and so of others. It is not then

alone sufficient in demonstration that its propositions are true,

nor that they are immediate, or such as inherit an evidence more

illustrious than the certainty of proof; but, besides all these, it

is necessary they should be made peculiar by a limitation of their

comprehensive nature to some particular subject. It is on this

account that no one esteems the quadrature of Bryso[24], a geometrical

demonstration, since he uses a principle which, although true,

is entirely common. Previous to his demonstration he supposes two

squares described, the one circumscribing the circle, which will be

consequently greater; the other inscribed, which will be consequently

less than the given circle. Hence, because the circle is a medium

between the two given squares, let a mean square be found between

them, which is easily done from the principles of geometry; this mean

square, Bryso affirms, shall be equal to the given circle. In order to

prove this, he reasons after the following manner: those things which

compared with others without any respect, are either at the same time

greater, or at the same time less, are equal among themselves: the

circle and the mean square are, at the same time, greater than the

internal, and at the same time less than the external square; therefore

they are equal among themselves. This demonstration can never produce

science, because it is built only on one common principle, which may

with equal propriety be applied to numbers in arithmetic, and to times

in natural science. It is defective, therefore, because it assumes no

principle peculiar to the nature of the circle alone, but such a one as

is common to quantity in general.

13. It is likewise evident, that if the propositions be universal,

from which the demonstrative syllogism consists, the conclusion must

necessarily be eternal. For necessary propositions are eternal; but

from things necessary and eternal, necessary and eternal truth must

arise. There is no demonstration, therefore, of corruptible natures,

nor any science absolutely, but only by accident; because it is not

founded on that which is universal. For what confirmation can there

be of a conclusion, whose subject is dissoluble, and whose predicate

is neither always, nor simply, but only partially inherent? But as

there can be no demonstration, so likewise there can be no definition

of corruptible natures; because definition is either the principle of

demonstration, or demonstration differing in the position of terms, or

it is a certain conclusion of demonstration. It is the beginning of

demonstration, when it is either assumed for an immediate proposition,

or for a term in the proposition; as if any one should prove that man

is risible, because he is a rational animal. And it alone differs

in position from demonstration, as often as the definition is such

as contains the cause of its subjects existence. As the following:

an eclipse of the sun is a concealment of its light, through the

interposition of the moon between that luminary and the earth. For

the order of this definition being a little changed, passes into a

demonstration; thus,

The moon is subjected and opposed to the sun:

That which is subjected and opposed, conceals:

The moon, therefore, being subjected and opposed, conceals the

sun.

But _that_ definition is the conclusion of demonstration, which

extends to the material cause; as in the preceding instance, the

conclusion affirming that the subjection and opposition of the moon

conceals the sun, is a definition of an eclipse including the material

cause.

Again, we have already proved that all demonstration consists of

such principles as are prior in the nature of things; and from

hence we infer, that it is the business of no science to prove its

own principles, since they can no longer be called principles if

they require confirmation from any thing prior to themselves; for,

admitting this as necessary, an infinite series of proofs must ensue.

On the contrary, if this be not necessary, but things most known

and evident are admitted, these must be constituted the principles

of science. He who possesses a knowledge of these, and applies them

as mediums of demonstration, is better skilled in science, than he

who knows only posterior or mediate propositions, and demonstrates

from posterior principles. But here a doubt arises whether the first

principles of geometry, arithmetic, music, and of other arts, can ever

be demonstrated? Or shall we allow they are capable of proof, not by

that particular science which applies them as principles or causes

of its conclusions? If so, this will be the office of some superior

science,--which can be no other than the first philosophy, to whose

charge the task is committed; and whose universal embrace circumscribes

the whole circle of science, in the same manner as arithmetic

comprehends music, or geometry optics.--This is no other than that

celebrated _wisdom_ which merits the appellation of science in

a more simple, as well as in a more eminent degree than others: not,

indeed, that all causes are within its reach, but such only as are the

principal and the best, because no cause superior to them can ever be

found. Hence the difficulty of knowing whether we possess science or

not, from the difficulty of understanding whether it is founded on

peculiar or common principles; since it is necessary that both these

should be applied in the constitution of all real knowledge and science.

[25]Again, axioms differ from postulates in this:--they demand our

assent without any previous solicitation, from the illustrious

certainty they possess. Their truth may, indeed, be denied by external

speech, but never from internal connection. He who denies that

equal things shall remain from the subtraction of equal, dissents,

as Euripides says, with his tongue, and not with his heart. But

demonstration depends not on external speech, but on intellectual and

internal conviction; and hence, axioms derive all their authority

from intrinsic approbation, and not from public proclaim. For the

prompt decisions of the tongue are frequently dissonant from the

sentiments concealed in the secret recesses of the heart. Thus the

[26]geometrician does not speculate those lines which are the objects

of corporeal sight, but such as are exhibited by mental conception,

and of which the delineations on paper, or in the dust, are no more

than imperfect copies, notes, and resemblances. Thus, when he draws

a pedal line which is not pedal, or an equilateral triangle which is

not equilateral, we must pay no regard to the designations of the pen,

but solely attend to the intellection of the mind; for the property

demonstrated of some particular line, is in the conclusion applied

to one that is universal, and this true line could be no otherwise

signified to the learner than by a material description.

The certainty of axioms is, indeed, in a measure obvious to every one.

For what more evident than that nothing exists of which it is possible,

at the same time, to affirm and deny any circumstance of being? Indeed,

so illustrious and indubitable is the light of this axiom, that in any

demonstration we are ashamed to assign it the place of an assumption.

It would almost seem prolix and superfluous, since there is nothing

more manifest and certain; and yet there are cases in which it is

necessary to rank it among assumptions. And these take place whenever

the intention is to conclude the existence of something as true, and

of its opposite as false. Thus, for instance, in the demonstration

that the world is finite, we assume this principle, and then reason as

follows:

Bound and infinite cannot be at the same time affirmed and denied

of any _body_:

The world is a body:

Therefore the world is not at the same time finite and infinite.

And in this genus of demonstration, the major proposition ought always

to assimilate with the conclusion. But the above axiom is not the

only one obvious, for the following possesses equal certainty; that

of every thing which exists, either affirmation or negation is true.

This axiom is of great use in demonstrations leading to an absurdity;

for he who demonstrates the impossibility of any opposite assertion,

necessarily establishes his own. Hence it is we affirm that the

diameter of a square is either commensurable or incommensurable with

its side; and this general principle is accommodated, and, as it were,

descends into its proper matter as often as that which it possesses of

universal is contracted to a certain genus; for, as we have previously

observed, common principles are not admitted in demonstration without

any restriction; but then only when their general nature is limited to

some particular subject, by which they become peculiar and apposite.

14. [27]Wisdom, or the first philosophy and logic, agree in not using

axioms after the same manner as other arts; but on the contrary,

they confirm and establish their certainty, though with this

difference, that the logician reasons only from probabilities, but the

metaphysician from the highest certainty and evidence. Besides, we do

not rank logic in the order of the sciences, because it is destitute

of some determinate genus or subject, as it is neither conversant about

lines, nor numbers, nor proportions. And its chief concern is about

apparent properties, and not such as are essential to a subject.

Hence, in logical disquisitions, we confidently employ interrogations,

as equally subservient to the affirmation or negation of an opinion:--a

method utterly impracticable, if we only employed those principles

which are universally acknowledged; since it is impossible of the same

thing to prove contrary properties,--as of the soul, that it is mortal

and immortal; but he who demonstrates, assumes one definite part of a

question, because his purpose is not to interrogate, but to trace out

the latent paths of truth. And hence, if any one affirms that the soul

is moved, and immediately after denies it, he is no longer a subject

worthy the exercise of our discursive and reasoning powers.

Again, it may so happen, that the same science at one time considers

_why_ a thing is, at another only explains its existence, or that

it exists, without considering the cause. Thus, the syllogism which

concludes by mediate propositions, demonstrates without assigning the

proper cause: but that which determines by immediate ones, in a great

measure explains the cause or reason of existence. Thus, he who infers

that trees do not breathe because they are not animals, reasons from a

mediate and secondary cause, because there are many animals, such as

insects, which exist without breathing: but he who infers this from

their want of lungs, demonstrates from the immediate and primary cause.

Thus, the following syllogism is a mediate one, or such as requires one

or more mediums to establish its certainty:

Every thing that is not an animal does not breathe;

A tree is not an animal;

Therefore a tree does not breathe.

Here the major proposition is evidently mediate, because we are still

to seek why that which is not an animal does not breathe, which the

following immediate syllogism solves.

Every thing that is not endued with lungs does not breathe;

Every thing that is not an animal is not endued with lungs; ergo,

Every thing that is not an animal does not breathe.

Again, the same science may demonstrate the existence of a thing,

or that it exists, and the cause of such existence as often as it

assigns two immediate reasons; but the one from the proper cause,

the other only from a sign. Thus, he who demonstrates the increase

of the moon, from the plenitude of her orb, infers the cause of such

increase; but on the contrary, he who collects the plenitude of her

orb from her increase, reasons only from a sign, and can alone declare

its existence. And, indeed, it often happens that the _cause_

and _sign_ reciprocate, so that as from the sign we advance to

the cause, demonstration from the cause frequently recurs to the

_sign_. Thus, from the breadth and firmness of the basis, we

collect the permanent duration of the pyramid; and from its extended

existence we infer the strength of its support. Whenever, then, the

argument originates from a sign, it gives evidence to the conclusion,

as from something more known than its cause. When it begins from the

cause, it proceeds from that which is first in the order of nature, to

that which is last, and reasons as from the proper principle of the

thing.

Sometimes the _cause_ and _sign_ do not reciprocate. Thus,

although wherever there is smoke, we infer the existence of fire; yet

we cannot infer, that wherever there is fire smoke exists. Thus, from

the palace and the picture we collect the existence of the architect

and painter; but the last may exist without the first;--the living

architect without the actual palace; and the living painter without

the energies of his art. And thus it is that the cause is illustrated

by its sign; but not always the sign by its cause.

Hence then, as all causes do not reciprocate with their effects; so

neither is it always causes and effects which do reciprocate: because

a multitude of signs, mutually inferring each other, may accompany a

certain cause. Thus, the signs which attend the causes of a fever, are

a quick pulsation of the artery, and an intense heat: and these signs

mutually assert each other; but no syllogism can be composed from

either expressing the _why_, but only simply _that_ the other

exists.

15. We now propose to consider the mode in which the two preceding

demonstrations are distributed in different sciences. When sciences

then are so related, that the one is dependent on the other, as optics

on geometry, navigation on astronomy, and music composed by the

arbitration of the ear, on that which consists in the knowledge of

mathematical proportions: in this case, the demonstration of simple

existence, or _that_ they _exist_, pertains to the science of

sensibles; but the demonstration _why_ they _exist_ to the

science which is speculative and mathematical.

Thus the mathematician speculates the causes of a certain sensible

effect, without considering its actual existence; for the contemplation

of universals excludes the knowledge of particulars; and he whose

intellectual eye is fixed on that which is general and comprehensive,

will think but little of that which is sensible and singular. Thus,

by mathematics we may learn the responsive harmony of the _last

chord_, and its consonance with the _mean_; but we cannot

perceive this concord, if unaccustomed to the practice of the musical

art. In fine, those sciences which are more of a mathematical nature,

I mean such as are more amply conversant with the inspection of

things, considering their forms abstracted from every material subject,

always demonstrate the _why_; and such is geometry in respect

of optics. Thus geometry considers only such things as are peculiar

to right-lines, independent of every sensible connection. For the

geometrician does not investigate a right-line as contained in stone or

brass; but considers it as entirely detached and unconnected with any

object of sense.

On the contrary, optics receives a right-line just as it is perceived

in a rule, or engraved in brass. And, indeed, in treating of some

particulars, natural science has the same relation to optics, as optics

to geometry. Thus, in considering the reason of the appearance of

the rainbow, the natural philosopher defines the bow to be an image

refracted from a certain cloud against the sun; but why it is endued

with such a form, and seen with such a colour, must be assigned by him

who is skilled in optics. There are, again, sciences, one of which is

not subordinate to the other, because founded on principles totally

different; yet, in some particulars they agree with the preceding.

Thus, to know that an orbicular wound is the most difficult of cure,

belongs to the physician; but to know _why_, to the geometrician.

16. Of all syllogistic figures, the first is the best adapted to

science, since the arithmetician, geometrician, and lastly all those

who demonstrate any effect from its proper cause, fabricate their

reasonings according to this figure. For the middle figure is seldom

used, because only adapted to a few occasions: and since the knowledge

of the _why_ is of all others the most important, which is alone

obtained by this figure: hence, in the pursuit of science, it is

always preferred before the rest. Besides, it is equally accommodated

to the knowledge of final causes; to which it alone tends: for it

composes definitions from words universal, and affirmative. In the

second figure, a complex negative is conceived; and in the last, a

particular one. Add to this, that mediate propositions are no other

ways reducible to immediate ones than by this figure, in which the

mediate proposition tends, by a continued series, to that which is

immediate. But the second does not conclude affirmatively, nor the

last universally; from whence it appears, that a mediate proposition

can never become immediate by these figures: not that all affirmative

propositions are immediate ones, since some negatives are of this

kind; for all propositions are equally immediate, which cannot be

confirmed by syllogism; and such are those negatives, of whose terms it

is impossible any genus can be affirmed. Thus the proposition, _no

substance is quality_, is an immediate negative of this kind, whose

terms are two of the most universal genera of things.

Again, as we have frequently affirmed that he who demonstrates, always

assumes such things as are essentially predicated; but that he who

argues dialectically or topically, not always, but generally assumes

such as are accidentally predicated, and which appear more probable and

known than such as are essentially inherent; it is proper we should

define what is meant by accidental predication; or something predicated

by means of another. Indeed, the term has a diffuse signification:

for, first, a body is said to be white by something else, because by

its superficies; and in this manner vines are white, because their

branches are white. Thus, if accident be predicated of accident, it is

by means of another; as when we say the musician is fair; for the being

a musician is an accident of man, and the being fair of the musician:

and man is the subject of each. The predicate of substance is equally

accidental, when not included in the number of things substantially

inherent; as when we affirm of any particular man that he is red, or

black. But the predication is especially accidental, as often as, by

perverting the order of nature, substance is predicated of accident; as

when we say something white is an animal: for this assertion differs

from that other, _animal is white_. In the latter, the subject

_animal_ is neither inherent in another, nor subsists by another,

but has an essential existence. In the former, what is assumed as a

subject derives its existence from that of which it is the accident.

It is only dialectically, therefore, that we can argue from predicates

as probable and known without any distinction: but in demonstration,

all that are preposterous and accidental must be carefully avoided,

excepting such accidents as being essentially in a subject, admit of an

essential predication; and some of these we have enumerated before.

17. We are now entering on a disquisition neither ignoble nor useless:

it is this, whether the number of things predicated essentially of a

subject is finite, or whether things in a continued series run on to

infinity. For instance, let us suppose some ultimate subject, which

is not the predicate of any thing besides; and let _c_ represent such

a subject, of which _b_ is the first and immediate predicate; and in

the same manner _d_ of _b_, and _e_ of _d_: the query is, Whether or

not this extraction must necessarily stop, or will admit of an immense

progression, so that _f_ may be predicated of _e_, and _g_ of _f_,

and so on infinitely; the power of the predicates, which supplies the

common identity, still remaining inexhaustible and undiminished? The

second query is this, Supposing some general subject, which we call

_a_, of such a nature as to be no longer the subject of any farther

predication, but to be itself the supreme and primary predicate; and

supposing that it is immediately inherent in _f_, and _f_ in _e_,

and _e_ in _g_, whether or not the process must stop, or extend to

infinity, and no subject be found which is not directly predicable of

another? There is a remarkable difference in the two considerations;

for, in the former we enquire whether any ultimate subject can supply

an infinite ascent of predicates; in the latter, whether any first

predicate can exist in an infinite descending series of subjects. The

third question is, supposing two extremes constituted from a first

predicate and last subject, whether it is possible an infinite number

of mediums can intervene? And this is no other than to enquire whether

demonstrations admit of an infinite progression, so that whatever is

assumed in proof of another, must be proved itself? Or whether it

is not more agreeable to truth, that there should be some immediate

propositions and ultimate terms, whose discovery may give respite to

enquiry, and stay the elaborate process of demonstration? The same

question occurs in negatives. But that some of these are immediate, the

instance lately alledged sufficiently evinces. The solution of this

enquiry is not so difficult in subjects which mutually reciprocate;

for in these, when the ultimate subject is given, no one can doubt the

existence of their primary predicate; nor when the primary predicate

is admitted, can there be any doubt of the existence of some ultimate

subject. For, in things which mutually reciprocate, whatever is

enquired of the one, is immediately questioned of the other; and

wherever there is a last subject, there must be a first predicate;

for by the conversion of the ultimate subject you effect the primary

predicate.

Previous to the discussion of the first question, it is necessary to

know that infinite intermediates cannot intervene between two finite

terms in an ascending and descending series of predications. I call the

series ascending which rises to universals; but descending, which, by

a contrary process, stops at particulars. Thus, if any one admits that

_a_ is some first predicate, and _g_ some ultimate subject,

and should contend, that between these terms there may be infinite

mediums, he contradicts himself; since he who begins from _a_ in a

descending progression, will never, by this means, arrive at _g_;

and he who departs from _g_ in an ascending series, can never

finally rise to _a_. So that the extremes can be no longer finite,

as the hypothesis admitted. Indeed, the absurdity of such a supposition

is the same as to contend that between one and ten, an infinity of

numbers may exist; which is evidently impossible, because the discrete

nature of numbers excludes their actual existence in infinitum,

between any finite limits; since they can only become infinite from

their actual existence and precedence, and not from any dormant power

or capacity they possess: for between any two given numbers there is

nothing similar to number in capacity, which can ever become number

in energy; as in quantity continuous between any two points there are

always parts in capacity, which, whenever a proper agent is at hand,

become immediately actual. In like manner, he who admits the terms

finite, but believes that the mediums are infinite, asserts what is

impossible, since these logical predications are of the same discrete

nature with numbers themselves. Thus all the predicates which can exist

between Socrates and substance, must exist actually, or not at all; for

surely between these two terms, or periods, no predicate in capacity

can ever be supposed to subsist. If it be urged, that the capacity of

receiving these predicates exists between Socrates and substance, still

we reply, it is not that kind of capacity in which these predicates can

retain the most shadowy existence; out of which they can ever be called

forth into energy, as from some latent retreat; or into which they

can finally retire, when energy is no more. And hence we conclude it

impossible that infinite mediums can exist between any finite terms.

18. It now remains that we prove, first, by probable arguments,

and then by such as are demonstrative, that the extremes in any

series of predications are finite; and that an infinite progression

is impossible, not only in substantial predicates, but in such as

are accidental. For every thing predicated of another is either

essentially or accidentally inherent; and is predicated in a natural

or preposterous order. It is predicated according to nature, when

accident is declared of substance; contrary to nature, when substance

of accident. That essential predicates are finite, appears from hence,

because a contrary hypothesis excludes the existence of definition,

by admitting that all things are contained in some superior genus,

and acknowledge some farther definition; since it is impossible that

the definitions of genus can ever be circumscribed, while there is a

continual supply of other genera, which can never be known without

definition; for thus we shall never obtain either a beginning or an

end. But to define all things is not possible, because infinity can

never be absolved by the most unwearied progression. Predictions then,

of this kind, are always circumscribed by a certain number of terms,

which prevent their infinite process, and cause all the strength of

demonstration, and all the certainty of human knowledge. The same may

be proved in accidents; for such as are predicated of substance, are

either predicated as qualities or quantities, as relatives, or as

actions and passions; as expressive of some habit, or significant of

some place; or as connected with some time. Thus we say the wood is

white, the triangle is scalene; whiteness being accidental to the

wood, and scalenity to the triangle. It is therefore certain, that

every accident is predicated of substance; and it is no less certain

that the predicates of substance are finite, since they are all

included in the ten universal genera of things.

19. We have hitherto defended the impossibility of an infinite

progression of logical predicates and subjects, in a demonstrative

process, by such arguments as are dialectical and common: it now

remains that we adopt such as are peculiar and certain. Demonstrations,

then, are derived from affections essentially inherent in a subject;

and these are either such as take place in definitions of a subject,

as multitude and quantity, are essentially predicated of number; or,

secondly, accidents which are defined from their subjects, as imparity

by number. But the predication cannot, in either case, be extended to

infinity. For it is not necessary that in the same manner that imparity

is predicated of number, something else, suppose _c_, should be

predicated of imparity; and so imparity be contained in its definition,

similar to number in the definition of imparity. For in predications

of this kind, the terms are always assumed more contracted than their

subject; and at length, by a continued procession, must terminate in an

indivisible. Thus, as imparity is more contracted than number, _c_

must be more contracted than imparity. Hence, these predications either

finally stop, for the reasons we have assigned; or because whatever is

predicated of imparity, is necessarily predicated of number; so that

one thing as number would be actually contained in the definition of an

infinity of things; and so actual infinity must ensue, which is absurd.

Lastly, whatever is said to reside in the terms, must be allowed to

reside in the subject; so number must be applied in the definition

of every affection; and an infinite number of properties will be

essentially inherent in number; and number will inherit infinite

definitions. But affections essentially resident in a subject cannot

be infinite, because it is necessary they should exist in energy.

Thus, imparity cannot exist potentially in number; nor reason in man;

nor rotundity in a circle, because wherever these subjects have an

actual being, it is necessary these essential attributes should be

actually inherent. Again, in the definitions of a subject, an infinite

process is impossible, because from such an hypothesis nothing could

ever be defined; and thus it appears that neither can demonstrations

be infinitely extended, nor every thing admit of demonstration, an

opinion we have already noticed in the beginning of this section: for

if neither universally, nor in every proposition a middle term can

be assumed, but as soon as we arrive at immediate propositions, the

labour of investigation is finished, the possibility of demonstrating

every thing can no longer be defended; since it is proved above, that

by limiting the extremes, an infinite number of mediums is necessarily

excluded.

And thus, by taking away infinity from the reasoning art, we have

given a support to science, which the most vigorous efforts of subtle

sophistry can never finally subvert. We have set bounds to that

restless spirit of enquiry which wanders uncontrouled in the mind

unenlightened by science, by every where circumscribing its progress

within the limits of that which is most particular, and most universal,

a first predicate, and an ultimate subject: and finally, by asserting

that all the evidence of human knowledge results from the lustre

of primary and immediate principles, we have held up a steady and

permanent light, ever sufficient to direct our steps through the dark

mazes of ignorance and error, into the bright paths of certainty and

truth.

20. Let us next consider whether universal demonstration is preferable

to particular, or not. And first, in favour of particulars we may

say that their evidence is more exquisite and certain than that of

universals. Thus, the knowledge, from inspection, that Callias is a

rational animal, is superior to that acquired by a reasoning process

which infers his rationality, because every man is a rational animal.

By particular demonstration a thing is known as it is, by universal

only in common. Besides, particulars possess some solidity, universals

none: and the demonstration of things which have a real existence,

is more excellent than that of things which have none. And there are

no errors more frequent than those about universals; demonstration

considering them as things entirely abstracted from singulars. On

the contrary, particulars are usurped by the sight, grasped, as it

were, by the hand, and the general subject of every sense; so that

concerning these, demonstration affirms nothing false or inconstant.

But these reasons, however plausible, are easily confused. And,

first, the term essential is more closely connected with universals

than particulars. Thus the possession of three angles equal to two

right, is an affection more essential to the triangle itself, than

to one equilateral or scalene. Add too, that in the demonstration of

universals we always infer some property of a subject from its simple

existence, or because it is such a subject. Again, many affections are

contained in singulars assumed from no particular nature, but from

that which is universal; as rationality in Socrates, which is not

inferred from his existence as Socrates, but from his existence as man.

Farther, that demonstration is the more excellent which is derived

from the better cause: but an universal cause is more extended and

excellent than a particular one; since the arduous investigation of

the _why_ in any subject is stopt by the arrival at universals.

Thus, if we desire to know why the exterior angles of a triangle are

equal to four right ones, and it is answered, because the triangle

is isosceles; we again ask, But why because isosceles? And if it be

replied, because it is a triangle, we may again enquire, But why

because a triangle? To which we finally answer, because a triangle is a

right-lined figure; and here our enquiry rests at that universal idea

which embraces every preceding particular one, and is contained in no

other more general and comprehensive than itself. Add too, that the

demonstration of particulars is almost the demonstration of infinites;

of universals, the demonstration of finites.--We add farther,

_that_ demonstration is the best, which furnishes the mind with

the most ample knowledge; and this is alone the province of universals.

Again, the principles of science become immediate only in proportion

as the demonstration becomes universal; and he who knows universals,

knows particulars in capacity: but we cannot infer, that he who has

the best knowledge of particulars, knows any thing of universals.

Lastly, that which is universal, is the province of intellect and

reason, particulars are the offspring of sense; and hence we conclude

that universal demonstration exceeds particular both in dignity and

excellence, and is first in the nature of things, although last in the

progressions of the reasoning power.

Again, That affirmative demonstration is superior to negative, appears

from hence: the affirmative does not require the assistance of the

negative; but the negative cannot exist without the affirmative;

on which account, the demonstration composed from negatives alone,

is incapable of producing real evidence and conviction. Besides,

affirmation exceeds negation both in priority and simplicity of

existence.

Again, the demonstration which concludes _directly_, is better

than that which confirms a proposition by evincing the absurdity of

its contrary. The first proceeding in a regular order, establishes,

by a natural deduction, the truth which was first advanced. The

second taking a wider circuit, yet with the same intentions produces

a conclusion quite opposite to its apparent design. The one may be

compared to the open attack of a valiant and skilful soldier, who

expects the conquest of his enemy from strength and courage alone: the

progress of the other resembles the same soldier, uniting force with

stratagem, and advancing, by an irregular march, which his foe mistakes

for a retreat, but finds the secret cause of his destruction. The first

is simple and impromiscuous, as composed from propositions alone: the

second is compound and miscellaneous, calling in hypothesis to its

assistance.

21. One science is said to be prior to, and more certain than another

in many respects;--when the one reasons from primary causes, but the

other from such as are secondary:--when the one may be ranked in the

genera of intelligibles and universals; but the other in the genera of

sensibles and particulars. And such is the relation of arithmetic to

music; of geometry to optics; and lastly, of every superior to every

subordinate science. Again, this happens when the one reasons from

simple principles, the other from such as are complex and connected;

on which account arithmetic seems to possess greater certainty than

geometry. For the principle of arithmetic is unity; but of geometry

a point; and unity is without position, with which a point is always

connected. And in this manner geometry inherits greater evidence than

astronomy; for the one considers body simply, the other as connected

with a circular motion. The science is called one which contemplates

actions belonging to one genus: the genus is one which possesses the

same first principles; and hence geometry and stereometry form one

science. On the contrary, the sciences are called different which have

different principles, such as geometry and optics; the latter of which

does not originate from the principles of the former.

Again, the same thing may admit of many demonstrations, and may be

known from many mediums: at one time from the application of such

as are congenial: at another, from those of a different order or

genus. From congenials, as when we demonstrate that the plantain is a

substance, first, by the medium of a tree, and then by the medium of a

plant, thus:

Every tree is a substance;

The plantain is a tree:

Therefore the plantain is a substance. And again,

Every plant is a substance:

The plantain is a plant:

Therefore the plantain is a substance.

We demonstrate, from mediums, of a following order or genus, as when we

prove man to be a substance, at one one time from his being rational,

at another from his being a biped; and these mediums, in part, mutually

contain each other.

22. Fortuitous events can never, in any science, become the subject

of demonstration; since they are neither limited by necessity, nor

admit the arrangement of syllogism. Indeed, so far from obtaining

a necessary, they do not possess a frequent existence, but every

syllogism is composed from one or other of these.

Again, science is not the business of sense, since that which is

universal is the object of perception in particulars themselves. For

the object of sight is colour in general, and not this particular

colour: the object of hearing is sound in general, and not any

particular sound; and, on this account we see or hear not only this

or that colour or sound, but likewise every other which falls under

the cognizance of these senses. Hence, if it were possible for any

one to discern by his sight, the equality of the three angles of some

particular triangle to two right, he would not by this means possess a

demonstration of the conclusion which affirms this to be the property

of every triangle; but his knowledge would extend no farther than the

triangle he inspects. Thus too, if we could perceive an eclipse of

the moon to arise from the interposition of the earth, we could not

universally conclude that this is the cause of every eclipse, but

only of the particular one we behold. For the explication of causes

extends to universals; and comprehends not only the knowledge of one

particular defect of the moon, but simply of every eclipse; since the

interposition of the earth is not so much the cause of any present

eclipse, as of all which can possibly exist in every age. Whenever,

then, the cause is universal, the knowledge of any effect deduced from

such a cause is, in every respect, superior to the evidence arising

from the perceptions of sense. It is likewise more excellent than

the apprehension which subsists independent of the proper cause; as

if any one should give absolute credit to the proposition, that the

three angles of a triangle are equal to two right, without a previous

conviction that the external angle of a triangle, is equal to the two

interior opposite ones; and without applying this last proposition as

the cause of the first. The comprehension, then, which is conjoined

with the proper cause, far exceeds the strongest evidence of sense.

But perhaps it may be said that science consists in sense, because the

science of any particular, fails from a defect of the sense by which

it is apprehended. To this we reply, that science, indeed, is not

acquired without the assistance of sense, but it does not follow from

hence, that _to perceive_ is _to know_; because the object

of science is that which is universal; but of sense, that which is

particular. Thus, if we could see light penetrating the pores of glass

(on the atomical hypothesis) the cause why it illuminates would be

manifest from sensible inspection as the means, and from the universal

apprehension of science, by which we should understand this to be

universally true.

Again, the principles of all sciences cannot be the same neither

considered as remote or proximate. Not considered as proximate, because

the principles always correspond to the demonstrated conclusions; but

these are not the same, since they are often generically different;

and consequently the propositions from which they result must be

derived from discordant genera. But propositions consist of such

things as essentially exist; and hence we infer, that the principles

of geometry are essentially distinguished from those of arithmetic,

that they cannot admit of reciprocal accommodation, so that the one may

be predicated, or become the subject of the other, and that the one

can never be subservient as a medium to the other. Again, common and

first principles are not applied in every science; such as this, that

every thing must either be affirmed or denied. Nor can any thing be

proved by their assistance alone, but as often as these are required

in demonstration, other principles more proximate and peculiar to the

given proposition, must always be adopted. Again, axioms universally

conceived, cannot be assumed in syllogism, but they must be contracted,

as it were, to some subject genus. Of this kind is that common

axiom, that as often as any four quantities are proportionable, by

permutation, or changing the order of the terms, the same ratio will

result. For the arts apply this axiom in a restricted sense; geometry,

by considering the relatives as four magnitudes, and arithmetic as four

numbers; but the natural philosopher, by adapting the comparison to

four motions, or four times. Besides, if the principles of all sciences

were the same, it is necessary they should be comprehended by some

certain number, similar to the limitation of the elements: but every

science is capable of immense increase from the many different modes

of amplification the conclusions will admit; and consequently it is

requisite to establish a correspondent number of proper principles;

for such as are common cannot be alone sufficient. Lastly, if the same

principles accord with every science, it follows, that any thing may

be demonstrated from such principles: but the certainty of geometrical

conclusions cannot be established from the principles of music; and

from hence it follows, that although the principles of every science

are not the same, they do not possess an entire diversity, nor yet an

absolute affinity of nature.

23. There is a remarkable difference between science and opinion.

Whatever is the subject of science must have a necessary existence; on

the contrary, opinion is conversant with things liable to mutation and

decay. Again, as science depends on necessary propositions for support;

so opinion on such as possess only a possibility of existence; and so

there is one mode of approbation in subjects of opinion, and another

in those of science. Hence science is distinguished from opinion by

two discriminations, the one arising from their subjects, the other

from the mode of approbation. That opinion is conversant with things

possible or contingent, we may learn from hence; contingencies cannot

belong to science, because their existence is not necessary; nor to

intellect, or that principle of science by which its terms are known;

nor to the apprehension or belief of immediate propositions, called

indemonstrable science. Hence, if every habit by which truth is known,

is either science, or intellect, or opinion, it remains that opinion

alone consists of things which are, indeed, true; but not necessary.

It is, therefore, inconstant and unstable, from the mutable nature of

its subjects. Besides, no one thinks he possesses an opinion of things

which he believes to have a necessary existence, so that they cannot be

otherwise than they are; but to such conviction he properly gives the

name of knowledge, and to its contrary the name of opinion.

Again, the same thing from the same propositions may at one time

become the subject of knowledge, at another, of opinion; and this

happens according to the different formation of the syllogism which

the propositions compose; whether reasoning from the proper cause it

explains the _why_, or only simply declares a thing exists. Hence

a doubt arises why opinions of this kind may not be called science,

since both the subjects and propositions are the same? The solution is

obvious. If it is believed that the propositions cannot be otherwise

than they are, or that they have a necessary existence, such an assent

of the mind is not opinion, but science; because things which inherit

an essential existence are the ornaments of science alone. On the

contrary, if we are convinced that the propositions are true, but

at the same time not necessary, such conviction is not science, but

opinion. Hence, it is impossible that science and opinion can be the

same, since they vary in their definition and mode of approbation, and

in a different manner demand our belief. Similar to this, although

it may happen that of the same thing a true and a false opinion may

arise, it will not therefore follow, that true and false opinions

are the same. For that which is firm and constant can never be the

same with that which is mutable and frail; and that which is always

true must be essentially different from that which may be changed into

false. By the power of habit indeed in different men, the same thing

may be comprehended by opinion and science. Thus it was opinion in

Epicurus when he said that the sun was eclipsed by the moon passing

under its orb, because he thought it might otherwise happen, and that

the moon might be interposed without obscuring the light of the sun.

It was science in Hipparchus, because he knew it as a necessary event.

But in the same mind, at the same time, and of the same thing, it is

impossible that science and opinion can exist. And thus much concerning

the difference of the two.

24. Lastly, sagacity is an acute and sudden apprehension of the medium,

or proper cause of a certain effect: as if any one, beholding the

moon, should in a moment conjecture the cause of the part opposite to

the sun being lucid, and the other parts obscure, because she derives

her splendor from the sun. Hence he is universally called acute and

sagacious, who, from the aspect or hearing of the extremes, can readily

perceive the medium which exists between them: as the term imports a

certain revolution of the conclusion into its first propositions, and,

as it were, a swift comprehension and continuation of the medium.