by G. J. Mattey

Book 1

*Of the UNDERSTANDING*

PART 2

*Of the ideas of space and time.*

Sect. 4. *Objections answer’d.*

**Context**

Sections 1 through 3 present the author’s system

regarding the ideas of space and time and their objects. In Section 1, the author argued that the ideas of space and time are not infinitely divisible, but must terminate in indivisible simple ideas. In Section 2, he applied this result to show that space and time themselves are not infinitely divisible. Section 3 is an attempt to demonstrate that the ideas of space and time are compound ideas that reflect the disposition of impressions of various sorts. In this and the following Sections, the author will attempt to rebut objections to his system.

**Background**

*Arnauld and Nicole*

The second knowledge by parts iswhen we consider a mode without paying attention to the substance or two modes which are united together in the same substance considering them each apart. This is what is done by the geometers who have taken as the object of their science body extended in length, breadth, and thickness. For in order to obtain a better knowledge of it, they have first applied themselves to the consideration of it in relation to one dimension alone, which islength; and they have then given to it the name ofline. They have afterwards considered it in respect to the two dimensions oflengthandbreadthand have called itsurface. And finally, considering all three dimensionslength,breadth, andthicknesstogether, they have called itsolidorbody.

Hence it may be seen how ridiculous is the argument of certain sceptics who would call in question the certitude of geometry, because it supplies lines and surfaces which are not in nature; for the geometers do not suppose that there are lines without breadth or surfaces without depth,—they suppose only that we are able to consider length, without paying attention to breadth; and this is indubitable, as when we measure the distance from one town to another we measure only the length of the road, without troubling ourselves with its breadth. (Chapter V, p. 46)

*Bayle*

Note G to the article Zeno

Bayle’s *Historical and Critical Dictionary* (1697) defends Zeno’s thesis that motion is impossible with new arguments to the effect that extension does not exist, in which case there is no motion. The basic argument is that if extension exists, it is composed of either mathematical points, physical points, or infinitely divisible parts. But none of these alternatives is viable, so extension does not exist. Bayle points out that the Schoolmen

acknowledge that the strongest alternative is that of infinitely divisible parts.

The Schoolman have armed [the hypothesis of infinite divisibility] from head to toe with all the distinctions that their great leisure was able to allow them to invent. But this served only to furnish their disciples with a jargon to use in public disputation, so that their relatives would not have the mortification of seeing them silent. A father or a brother goes home much happier when the student distinguishes between theTwo specific arguments given are that infinite divisibility would make contiguity impossible and it would allow for the inter-penetration of objects. According to the first argument, if objects are extended, then their parts are in immediate contact with one another. For any two parts of space are separated from each other by infinitely many parts. According to the second argument, extension does not allow the inter-penetration of the parts of two objects. But take any two objects that are supposed to be in contact with each other, like a cannon ball resting on a table. Pick any allegedcategormaticaland thesyncategormaticalinfinite, between thecommunicatingandnoncommunicatingparts,proportionalandaliquot,than if he had made no response.

point of contactA between the two and consider the part A of one object, say the cannon ball, which is touching the other, say the table, at point B. Bayle claims that it follows that

they mutually touch each other according to their depth, and consequently they penetrate one another.One consequence of the arguments against extension is that it must be treated existing only ideally, in the mind, just as do the objects of geometry, according to the mathematicians.

*Barrow*

Isaac Barrow’s 1734 *The Usefulness of Mathematical Learning Explained and Demonstrated* contains an account of the relation of equality of figures in geometry in terms of the relation of contiguity. The Supposition of [

(page 185). Contiguity itself is defined in three ways, the first of which is relevant to the present Section.
*Congruity*] is as it were the chief Pillar and principal Bulwark of all the Mathematics. For from hence, in my Opinion, is taken the formal Reason of Equality which is all in all in the Mathematics

Congruityis wont to be described byOccupation,Possession, orRepletionof the samePlaceorSpace:which may be conceived to done three ways,viz. byApplication,Succession, orMental Penetration. (1) ByApplication, when one Magnitude is conceived to be so laid upon or applied to another as immediately to touch that other with all its Parts, and no where recede or be separated from it: As when a Measure is applied to, and coextended with the measured (as a Yard-Line to the Selvage of a Web; or assigned Right Lines to the Surface the Earth) in whichApplicationall the Parts of one Longitude do exactly answer to and immediately touch the Parts of the other, and consequently these are congruous to one another after this manner. (page 189)

*Chambers*

Cyclopædia,containing an article entitled

Judgment. Chambers argued that assessments such as those of the relative size of bodies are merely perceptions of a certain kind. Perception proper has one object in view, while judgment relates two or more objects. An example of judgment is the comparison of sizes.

Thus when we judge that the Sun is greater than the Moon, and finding the Idea of the Sun greater than that of the Moon, the Will perfectly acquiesces in that Perception, nor puts the mind upon any further inquiry.

*Butler*

In Appendix I, Of Personal Identity,

to his 1736 *The Analogy of Religion,* Bishop Joseph Butler noted that the difficulty in defining personal identity is the same as that of defining equality, and the solution to the difficulty is the same.

Now, when it is asked wherein personal identity consists, the answer should be the same as if it were asked, wherein consists similitude or equality; that all attempts to define, would but perplex it. Yet there is no difficulty at all in ascertaining the idea. For as, upon two triangles being compared or viewed together, there arises to the mind the idea of similitude; or upon twice two and four, the idea of equality; so likewise upon comparing the consciousness of one's self, or one's own existence in any two moments, there as immediately arises to the mind the idea of personal identity. And as the two former comparisons not only give the idea of similitude and equality, but also shows us, that two triangles are like, and twice two and four are equal; so the latter comparison not only gives us the idea of personal identity but also shows us the identity of ourselves in those two moments; the present, suppose, and that immediately past; or the present, and that a month, a year, or twenty years past. Or, in other words, by reflecting upon that which is myself now, and that which was myself twenty years ago, I discern they are not two, but one and the same self.This view is endorsed by the author in paragraph 22 below.

**The Treatise**

1. The author describes his system concerning space and time

as consisting of two intimately connected parts. The chain of reasoning leading to the first part proceeds as follows. The capacity of the human mind being finite, there is no idea of extension or duration that consists of infinitely many parts or ideas of lesser extension or duration. The parts of all extensions and durations are finitely many, simple and indivisible. [Given that what is conceivable is possible,] it is possible that space and time themselves consist of finite, simple, indivisible parts. On the other hand, it is impossible that the parts of extension and duration are infinitely divisible. Therefore, space and time (extension and duration) consist of finitely many simple, indivisible parts.

2. The second part of the system is said to follow from the first part. The parts of space and time are indivisible, and unless they were filled

with something
real and existent,

they would be nothing, non-entities. Space and time themselves [which are composites of these unextended or momentary parts] consequently are not ideas separate and distinct [from their parts], but can only be merely those of the manner, or order, in which objects exist.

The author describes this conclusion in other words

as the claim that it is impossible to conceive either a vacuum and extension without matter, or a time, when there was no succession or change in real existence.

[The alleged idea of either a vacuum or a time without change would not be a copy of an impression or a representation of the disposition of impressions, since by their very nature they are empty of objects or of change.] Because these two parts of the system are connected intimately, the objections to them will be dealt with together, beginning [in this Section] with the objections to the first part, which claims that extension is only finitely divisible.

3. The first objection to be considered, far from undermining the system, is more properly described as proving the connexion and dependence of one part upon the other.

It is often claimed in the schools

that it is absurd to allow the existence of a mathematical point, because it is a non-entity that can never by its conjunction with others form a real existence.

Mathematical points need not be invoked if matter is infinitely divisible, since the division need never terminate in a point. The author allows that this reasoning woul’d be perfectly decisive

of the only two alternatives were the infinite divisibility of matter and the existence of [invisible and intangible] mathematical points. However, there is a third alternative or medium,

which is mathematical points that are endowed with color or solidity. Because both the infinite divisibility of matter and the existence of [invisible and intangible] mathematical points are absurd, the truth and reality of this medium

is demonstrated. Another medium is a system of [extended] physical points. This is also absurd, in that real extension always contains parts which are different from one another. As such, they are distinguishable and separable from one another by the imagination [in which case they cannot be designated as points, which are indivisible]. [The reason the answere to this objection cements the connection of the two parts of the system is that it shows that indivisible parts of space are real existents, which rules out the possibility of parts of space that are not real existences, i.e., empty spaces.]

4. The second objection concerns an alleged consequence of the doctrine of visible and tangible mathematical points. Such a point is perfectly simple and therefore lacking in any parts. Thus, it is not possible for one part of a point to touch another point. If there can be no parts of points to touch other points, if two points in fact touch each other, one would have to penetrate

the other, that is, touch it intimately, and in its whole essence.

The author also uses the scholastic Latin expression ‘*secundum se tota & totaliter*, or according to itself, totally and completely.

This relation is the very definition of penetration.

The argument continues by claiming that this kind of penetration is impossible. So, [if a mathematical a point must penetrate another to touch it,] it cannot possibly exist.

5. The objection depends on the definition of penetration and would lose its force if a juster

definition is used—and this is just what the author proposes. The proper notion of penetration is that of two perfectly solid bodies (containing no void within their circumference

) were to approach each other and become united such that the resulting composite body is no more extended than either of the two bodies making it up. [Thus, two bodies one inch in diameter would, upon touching, comprise a body that is itself one inch in diameter.] This kind of penetration is impossible: the result of such an interaction could only be the annihilation of one of the two bodies, though we might not be able to tell which was preserved and which annihilated. Before the approach we have the idea of two bodies. After it we have the idea only of one.

We cannot form a notion of two bodies existing in the same place at the same time.

6. So we must ask ourselves what would happen if two colored or solid points were to approach each other and touch. Surely we would not think that there was penetration, and that one of the two bodies is annihilated. Rather, it is evident that the result would be a compound body consisting of two existing, distinguishable, bodies lying side-by side. The author proposes a thought-experiment. Suppose one point is red and the other blue are in this situation. If they do not lie side-by-side without one being annihilated, we must ask questions we cannot answer, such as which of the two colored points survives and which perishes, or whether a new color is produced.

7. The problem that primarily makes these considerations so difficult to resolve is the fact that we are dealing with the thought of very minute objects, which are not well-grasped by our infirm and unsteady senses and imagination. An example is given to illustrate how unsteady the senses and imagination are. We are asked to put a spot of ink on a piece of paper and to begin moving away from it, until the spot can no longer be seen. When we move toward it again, we begin to pick it up and then lose sight of it. Upon still nearer approach, it is always visible. Then we get a more forceful impression of its color, without its appearing any larger. Finally, it begins to look really extended, at which time it is difficult for the imagination to break it into its component parts, because of the uneasiness it finds in the conception of such a minute object as a single point.

This weakness of the imagination affects most of our reasonings on the present subject, and makes it almost impossible to answer in an intelligible manner, and in proper expressions, many questions which may arise concerning it.

8. Many objections to the doctrine of perceptible mathematical points have been drawn from mathematics. The author holds that in fact mathematics seems at first sight to favor this doctrine. While it may be that there are mathematical demonstrations that conflict with it, the definitions in mathematics do not. My present business then must be to defend the definitions, and refute the demonstrations.

9. The following definitions hold in geometry:

- A surface is length and breadth without depth,
- A line is length without breadth or depth,
- A point has neither length, breadth or depth.

intelligibleis the supposition that extension is composed of

indivisible points, or atoms.This is the only conceivable way that an object can exist without any dimensions at all.

10. There are two distinct rebuttals to this argument, neither of which the author finds to be satisfactory.

The first objection is that the objects of geometry defined above exist only in the mind and not in extra-mental reality. They never did exist; for no one will pretend to draw a line or make a surface entirely conformable to this definition. They never can exist; for we may produce demonstrations from these very ideas to prove, that they are impossible.

[Thus, the fact that the only way for surfaces, lines and points to exist is if there are atoms does not imply that there must be such atoms.]

11. The author finds this reasoning to be the most absurd and contradictory

as can be imagined. He appeals to the principle that whatever idea can be conceived clearly and distinctly must at least possibly exist. Anyone who tries to prove the impossibility of the existence of what is clearly and distinctly perceives can only by this means show that he did not conceive it clearly and distinctly. In this way he contradicts himself by beginning with a clear idea and then giving an argument whose consequence is that the idea was not clear. ’Tis in vain to search for a contradiction in any thing that is distinctly conceiv’d by the mind.

If an idea implies a contradiction, it cannot be conceived clearly.

12. There is no alternative possibility (medium

) besides those of allowing that indivisible points are at least possible and denying that we have a clear and distinct idea of them. This fact gives rise to a second objection to indivisible points. It is granted by some that we are not able to conceive objects that meet the definitions given above, but it is held at the same time that we can think about length without breadth, etc.

[Footnote. Reference is made to *The Art of Thinking*, the relevant portion of which is described above.]

This is done by a process of an abstraction without a separation.

An example is the thought of the length of distance between two cities without taking into account its breadth. Although both in the mind and in existing objects length and breadth are inseparable, we can consider objects partially and make what has been described earlier as a

[See Part 1, Section 7, paragraph 17. For example, we may consider a globe with respect either to its color or its shape, though the two are not separable in either the globe itself or the idea of it.]*distinction of reason*.

13. The first absurdity found by the author in this reasoning is his argument from Section 1 concerning the consequences of denying that extension has minimum parts. To be able to conceive of the infinity of parts that the idea of any extension would then have, the mind would have to be infinite, which it is not. Here he will endeavour to find some new absurdities in this reasoning.

14. The author appeals to specific qualities of surfaces, lines and points. Each serves as the termination of an object with one more dimension. Thus, a surface terminates a solid, a line terminates a surface, and a point terminates a line. But for this to be conceivable, the idea of a point must be indivisible. Suppose you wish to conceive of a termination, say that of a line in a point. Apply your imagination to the idea of any candidate for the terminating object, and it can be divided into smaller parts, and this to infinity. The imagination thereby has no possibility of arriving at a concluding idea.

Whatever the number of fractional parts you arrive at, you are no closer to the termination than when you began. Every particle eludes the grasp by a new fraction; like quicksilver, when we endeavour to seize it.

Yet there must be a termination to the geometrical object, and the only way this is possible is if there are surfaces that cannot be divided according to depth, lines that cannot be divided according to breadth, and points that cannot be divided according to length.

15. The author cites the schoolmen

as recognizing the need for terminating points, and he notes that some of them went so far as to mix indivisible mathematical points in with the infinitely divisible parts of extension. [A]nd others eluded the force of this reasoning by a heap of unintelligible cavils and distinctions

[see the quotation from Bayle above]. Either way, the author’s position is vindicated. The first view is simply a surrender, and the second is an attempt to hide. And hiding acknowledges the superiority of the adversary just as much as does laying down one’s arms.

16. This reasoning vindicates the author’s claim in paragraph 8 above that the system of indivisible points conforms to the definitions of geometry, and that any alleged demonstration of its falsehood is not consistent with those definitions and hence destroys geometry altogether. On the one hand, there is no inconsistency between the definitions of the geometrical objects and the ideas of indivisible points. So these are possible objects of geometry. But on the other hand, there is an inconsistency between the definitions of the terminations of geometrical objects and the denial that there are indivisible geometrical points.

17. The arguments of the author take a new turn at this point. He claims that none of these demonstrations can have sufficient weight to establish such a principle, as this of infinite divisibility.

The reason is that no demonstration can be made about the properties of objects that are so small that we cannot have exact ideas about them. This has the consequence that the theses of geometry are not precisely true.

Specifically, geometrical proofs about proportions of quantity are rough, though they are just

as far as they go. The errors of geometry are never considerable; nor woul’d it err at all, did it not aspire to such an absolute perfection.

18. The author poses a question that he thinks will embarrass

geometricians of any sect,

and specifically whether they claim that extension is composed of indivisible points or of quantities that can be divided to infinity. The question is what they mean when they compare the size of lines or surfaces and pronounce one of them to be equal to, greater than, or less than the other.

19. The readiest and surest

answer to this question is available to anyone who defends the doctrine of indivisible points, though such mathematicians are few or even none. Their response would be that lines or surfaces are equal when the numbers of points in each are equal; and that as the proportion of numbers varies, the proportions of the lines and surfaces is also vary’d.

[This is now known as Hume’s Principle,

and it is the cornerstone of well-known contemporary versions of logicism, usually referred to as neo-logicism.

] The author notes that while this answer be just, as well as obvious,

it turns out to be useless and in fact never used in making comparison of the size of geometrical objects. The problem is that the visible or tangible points to be counted are too small and easily confused with one another that the mind cannot make exact comparisons based on them and thus cannot judge the proportions in question. For example, one cannot count the points in an inch and a foot and then compare the two numbers to determine that a foot is larger than an inch, for which reason we seldom or never consider this as the standard of equality or inequality.

20. The situation is worse for those who claim that extension is infinitely divisible, because it is not merely difficult, but quite impossible, to compare the number of points making up two quantities of extension. The reason is that any two quantities of extension have infinitely many parts, and, according to the author, there is no proper sense in which two infinite quantities are equal or unequal. [There remains a problem even given the contemporary standard for comparison of the cardinality of infinite numbers. Any infinitely divisible finite quantity is the size of the continuum, and as such, all finite quantities of extension would be equal.] So, the quantities cannot be compared on the basis of the number of their parts. An objection is that we can compare quantities on the basis of the number of their parts, as when we compare a foot and a yard with respect to the number of inches contained in each. But to do this, we must have a way of showing that what is deemed an inch in a foot is identical to what is deemed an inch in a yard. To do this, we would have to sub-divide an inch into identical quantities and show that the number of subdivisions is the same. But the problem arises anew with respect to these smaller quantities, and so on to infinity. The author concludes that it is evident, that at last we must fix some standard of equality different from an enumeration of the parts.

21. An alternative way of comparing the size of geometrical objects was described by Isaac Barrow.

[Footnote. Reference is made to Barrow’s mathematical lectures, the relevant portion of which is described above.]

The best definition of equality of two figures, on this view, is based on the notion of their congruity. Congruity, in turn, can be measured by superimposing one figure on another and determining whether all their parts correspond to and touch each other.

The author first notes that equality is a relation and not, strictly speaking, a property in the figures themselves.

The existence of the relation is based on a comparison of two objects in the imagination. To determine whether two figures are congruous, we must have a distinct notion

of their parts and the mutual contact of those parts. We cannot use the gross parts of the figures to determine congruity, [since we cannot judge whether these parts are exactly corresponding and touching]. So we must run up these parts to the greatest minuteness, which can possibly be conceiv’d.

However, the smallest conceivable parts of extended quantities are mathematical points. As a result, we have the same problem as noted in paragraph 19 above: the parts are too small to compare accurately. [The author in fact claims that this standard of equality is the same with that deriv’d from the equality of number of points.

However, the standard is not strictly the same in all cases. Two geometrical objects might have the same number of points, not all of which touch each other, as with a straight and curved line of the same length. Barrow’s notion of congruity was meant to apply to more than straight lines and plane surfaces. If applied to straight lines, then two lines with the same number of points are congruous.] The criterion of having the same number of points was shown to be just but of no use, so [w]e must therefore look to some other quarter for the solution of the present difficulty.

22. [Appendix. It has been maintained by many philosophers

that there is no standard of equality, but that we can get a proper notion of that relation by presenting two objects that are in fact equal. [One such philosopher is Bishop Butler, as noted above, with whom compare Ephriam Chambers, also noted above.] These philosophers claim that it is useless to give definitions of equality without perceiving equal objects, and that once we do perceive them, no definition is needed. The author agrees entirely with this view, claiming that the only useful notion of equality, or inequality, is deriv’d from the whole united appearance and the comparison of particular objects.

The reason is that] we evidently do ordinary make judgments of equality or inequality of quantities on a single view, without examining or comparing the number of their minute parts. Such judgments are not only common, but in may cases certain and infallible.

The superior size of a yard as compared to a foot can be doubted no more than can be those principles, which are the most clear and self-evident.

23. So, we judge, from the general appearances of the objects of the mind, the three proportions of equality, larger than, and smaller than. Although in many cases the judgments we make about these proportions are infallible,

in many others they are subject to doubt and error,

just as we are in other matters. We often use review and reflection

to correct our initial assessment that is based on mere appearance. Sometimes we can correct ourselves by juxtaposing the two objects, or by use of some common and invariable measure

applied to each one when they cannot practically be placed in physical contact with each other. And such measurements can be corrected further by using more exact instruments and taking greater care in their use.

24. Thus there are two methods of assessing equality, one looser

and the other stricter.

We judge equality in the first way by noting the general appearance

of two objects, and we judge it in the second way by comparing the objects to each other by juxtaposition and by use of common measures. When we get used to doing this, we form a mix’d notion of equality.

But we do not stop here, as sound reason convinces us that there are bodies

Moreover, *vastly* more minute than those, which appear to the senses.false reason

convinces us that there are bodies infinites smaller than these. We recognize that we cannot perceive these bodies nor have adequate measuring instruments. As a result, we form an imaginary

standard for comparing them in terms of equality. The standard must be imaginary because we cannot see the bodies, and we cannot make corrective measurements: i.e., we cannot juxtapose the bodies, nor have we a common measure of them. For this reason the standard is a mere fiction of the mind, and useless as well as incomprehensible.

Nonetheless, the creation of the fiction is natural, as are other fictions which result from the continuation of the use of a procedure that no longer applies to the present subject. A very conspicuous example is the division of time. We have inexact measures of the parts of time, even less exact as those we have for measuring extension. Still, whatever corrective measures we have lead us to form an obscure and implicit notion of a perfect and entire equality.

Other examples are given. A musician may through practice and correction get better and better at discerning notes, then fancying himself to be able to judge the perfect correspondence of two notes an octave apart. Painters think themselves able to judge equality of colors and mechanics equality of motions. To the one,

*light* and *shade*; to the other *swift* and *slow* are imagin’d to be capable of an exact comparison and equality beyond the judgment of the senses.

25. The same reasoning [regarding the relations of equality and inequality] may be applied to the distinction between curved and straight (right

) lines. Although nothing could be easier than to form the ideas of the two based on the ideas of objects we get from impressions of the senses, we cannot give definitions that will allow them to be distinguished with precision. When we draw a line on paper, there is a certain order of its parts that gives it the shape it has, but this order is not visible: only the united appearance

is observable. So, if we allow that the line is composed of indivisible points, our situation is that the points are unknown, and we can only form a distant notion of some unknown standard to these objects.

Such a distant notion cannot be formed at all on the hypothesis of the infinite divisibility of the line, and all we can have for a rule to determine whether it is curved or straight is its general appearance. This failure to provide an adequate definition of the two notions and to provide a standard for distinguishing them does not stop us from correcting the first appearance by a more accurate consideration, and by a comparison with some rule, of whose rectitude from repeated trials we have a greater assurance.

This process of correction, as well as the tendency of the mind to project its methods beyond the limits established by reason, allows us to form the loose idea of a perfect standard to these figures, without being able to explain or comprehend it.

26. A proposed definition of a straight line is given by mathematicians:

The author raises two objections to this proposed definitions. The first objection is *it is the shortest way betwixt two points*.that this is more properly the discovery of one of the properties of a right line, than a just definition of it.

Anyone who thinks of a straight line does not think of this definition, but rather thinks of a particular appearance,

discovering by accident the property that it is the shortest distance between two points. Moreover, one can understand what a straight line is by considering a single one. But the alleged definition involves a comparison between a straight line and other lines, the straight line being the one that is least extended. A further point along these lines is that the alleged definition is commonly accepted as a maxim.

If the idea of a straight line just is the idea of the shortest distance between two points, then asserting the maxim would be as absurd as asserting that the shortest way is always the shortest.

27. The second objection is based on the claim that has already been defended by the author: that we have no precise idea

when it comes to the notions of equality and inequality, or different lengths, and our idea of the difference between a straight and curved line is at least as imprecise. So we cannot build a precise standard for a straight line on that of comparative lengths of lines. An exact idea can never be built on such as are loose and undeterminate.

28. The considerations brought to bear against the precision of the idea of a straight line apply as well to that of a plane [or flat] surface. The only means we have of distinguishing a flat surface from a curved one is by its general appearance.

Mathematicians try to define a flat surface in terms of the flowing of a straight line, but this attempt is in vain. We do not form our idea of a surface in this way, any more than we form our idea of an ellipse in terms of a section of a cone. Moreover, the idea of a straight line is not any more precise that the idea of a flat surface. Further, a straight line could flow irregularly [perhaps in a wavy path], and so to insure that it describes a plane, its flow must be fixed by two straight lines, along which the first one flows. The problem here is that straight lines must be parallel to each other and on the same plane. But as this is an attempt to define a plane, the definition would be circular.

29. At this point, the author summarizes his results to date. The most essential ideas in geometry, those of equality and inequality, of straight lines, and of planes, are far from being exact and determinate, according to the common method of conceiving them.

It is not only the case that we cannot decide in dubious cases when two geometrical objects are equal or not, when a line is straight or not, or when a surface is flat or not, but we cannot form a firm and invariable

idea of equality and inequality, or of the properties of straightness and flatness. We can appeal only to the weak and fallible judgment, which we make concerning the appearances of the objects, and correct by a compass or common measure.

Any alleged corrective devices beyond these are useless or imaginary.

It is in vain that we appeal to the supposition of a God who could by virtue of his omnipotence produce an absolutely straight line. We cannot devise a standard in these matters except by the use of the senses and the imagination applied to general appearances. Since these faculties provide the standard, it is absurd to speak of perfection beyond it, because the standards themselves define what perfection is.

30. Given that the standards used in geometry are imprecise, the author challenges the mathematicians to justify infallibly not only their more intricate and obscure

theorems, but also their most vulgar and obvious principles.

For example, it is claimed that the intersection of two straight lines is not itself a line. Another example is that it impossible to trace two different straight lines between two points. The geometer might reply that the denial of these principles is obviously absurd, and repugnant to our clear ideas.

The author allows that this is the case when [in reference to the first case] two straight lines form a sensible angle to each other. But if the angle is imperceptible, there seems to be no absurdity in supposing that two lines might merge to form a single line. The author’s example is two lines whose angle of difference decreases by one inch every twenty leagues [sixty miles]. If it is denied that the two lines can share a common segment, it must be on the basis of some definition of straight lines that makes this impossible. The author suggests that the reason would be that no more than one point whose ordering are the essence of the two straight lines can be superimposed on each other. He responds by stating that he must inform

the objector, first that this response implies that lines are made up of indivisible points, which, perhaps, is more than you intend.

Secondly, the order of the points making up a line is not the basis of our conception of a straight line, [which is formed from general appearances]. And even if this were the standard, there is not any such firmness in our senses or imagination, as to determine whether such an order is violated or preserv’d.

It may appear that the two lines coincide, even if we try to correct the general appearance by all the means either practicable or imaginable.

31. [Appendix. The mathematician must always face the following dilemma. The standard of equality, or any other proportion

is either accurate and exact

or inaccurate.

The accurate standard is the enumeration of indivisible parts.

The inaccurate standard, which is the common one, is comparison of objects based on their general appearance, as it is corrected by measurement and juxtaposition of objects. If the mathematician adopts the accurate standard, it cannot be used, and moreover, it presupposes that extension is not infinitely divisible, a view that he tries to explode.

On the other hand, if he adopts the inaccurate standard, with its certain and infallible

principles, he will find them to be too coarse

to be used in the subtile inferences

that he commonly makes. Because the principles embodying the inaccurate standard are based on the imagination and the senses, their application cannot exceed, let alone contradict, them.]

32. The considerations that the author has brought forward may serve to open our eyes a little,

allowing us to see that the arguments for the infinite divisibility of extension can never be as forceful as other geometrical arguments. The reason why this argument fails, where all the others command our fullest assent and approbation

is also revealed by these considerations. In fact, it is more important to show the reason for this exception than to show that it is an exception, in which we regard all the mathematical arguments for infinite divisibility as utterly sophistical.

The reason it is an exception is that it purports to use ideas to prove that quantities are infinitely divisible, when ideas by their very nature are not infinitely divisible, a glaring absurdity

than which none greater can be imagined. Because this absurdity is vary glaring in itself,

any argument based on it is attended with a new absurdity and involves an evident contradiction.

33. An example of the absurdity of attempts to use ideas to demonstrate infinite divisibility is a set of arguments that involve the notion of the point of contact between the line describing a circle and a straight line [describing its tangent]. The use of diagrams on paper will not settle the question of whether there can be more than one point of contact between two lines, as these are only loose draughts,

which serve only to convey with greater facility certain ideas, which are the true foundation of all our reasoning.

With this, the author agrees, and he is content to settle the whole dispute on the bases of these ideas that are supposed to be the basis of our reasoning. So let the mathematician form ideas of a circle and a straight line. He must conceive of the contact to exist in a mathematical point or to consist of an overlap for some space

between the circle and the line. This poses a dilemma, with each side posing equal difficulties. Suppose in his imagination he conceives the circle and the straight line to coincide at one point. This is to allow that the idea of a mathematical point is possible, and therefore that mathematical points themselves are possible. On the other hand, if he can only imagine the two lines to overlap, then he must recognize that geometrical demonstrations are fallacious when they are carried beyond a certain degree of minuteness,

since he has other demonstrations to prove that the two cannot overlap. To put it another way, he can show that the idea of concurrence is incompatible with the ideas he has formed in his imagination of a circle and a straight line. But in imagining the overlap, he finds that the three ideas are inseparable

when he thinks about the meeting of a circle and a line.

**The Enquiry**

The system

of the *Treatise* is not discussed in the *Enquiry*, and the various defenses of it given here are likewise omitted. Geometry is treated there as a science as certain as those of arithmetic and algebra.

That the square of the hypothenuse is equal to the square of the two sides, is a proposition, which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is any where existent in the universe. Though there never were a circle or triangle in nature, the truths, demonstrated by Euclid, would for ever retain their certainty and evidence. (Section IV, Part I)

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