The use of organization to maintain the mind's attention, and on the usefulness of geometry

One must exercise great circumspection in the choice and use of the aids one can draw from the senses and passions to render oneself attentive to truth, because our passions and senses affect us too vividly and fill the mind’s capacity to such an extent that it often sees only its own sensations when it thinks it is discovering things in themselves. But it is not the same with the aids one can draw from the imagination: they make the mind attentive without uselessly occupying its capacity, and thus they wonderfully help to perceive objects clearly and distinctly, so that it is almost always advantageous to use them. But let us make this clear with a few examples.

A body is moved by two or several different causes toward two or several different sides; that these forces push it equally or unequally; that they increase or decrease incessantly, according to any known proportion one wishes. And the question is: what path must this body follow, where must it be at such a moment, what must its velocity be when it arrives at such a place, and other similar things.

1. From point A, which is supposed to be where this body begins to move, one must first draw the indefinite lines AB, AC, which form the angle BAC if they intersect; for AB and AC are direct and do not intersect when the movements they express are directly opposed. Thus one distinctly represents to the imagination — or, if one prefers, to the senses — the path this body would follow if there were only one of these forces pushing it toward either side A or B.

2. If the force moving this body toward B is equal to that moving it toward C, one must cut along the lines AB and AC the parts 1, 2, 3, 4, and I, II, III, IV, equally distant from A. If the force moving it toward B is double that moving it toward C, one cuts the parts on AB double those cut on AC. If this force is half, one cuts them half; if three times greater or smaller, one cuts them three times greater or smaller. The divisions of these lines further represent to the imagination the magnitude of the different forces moving this body, and at the same time the space they are capable of making it traverse.

3. One draws through these divisions parallels to AB and AC, in order to obtain the lines 1X, 2X, 3X, etc., equal to AI, AIII, etc., and IX, IIX, IIIX equal to Al, A2, A3, which express the spaces these forces are capable of making this body traverse; and through the intersections of these parallels one draws the line AXYE, which represents to the imagination: first, the true magnitude of the compound motion of this body, conceived as pushed at the same time toward B and toward C by two different forces in such a proportion; second, the path it must follow; finally, all the places where it must be at a determined time; so that this line serves not only to sustain the mind’s vision in the search for all the truths one wishes to discover on the proposed question, but even represents its resolution in a sensible and convincing manner.

First, this line AXYE expresses the true magnitude of the compound motion; for one sees sensibly that if the forces producing it can each advance this body by one foot in one minute, its compound motion will be two feet in one minute if the component motions agree perfectly; for in that case, it suffices to add AB to AC, because the forces of the component motions are entirely employed in forming the compound motion; and if these motions cannot entirely agree, the compound AE will be greater than one of the components AB or AC by the line YE. But if these motions proceed along two lines forming the angle CAB of 120 degrees, the compound will be equal to each of the equal components. Finally, if these motions are entirely opposed, the compound will be zero, because the forces of the component motions being equal, they balance each other.

Second, this line AXYE represents to the imagination the path the body must follow, and one sees sensibly in what proportion it advances more on one side than on the other. One also sees that all compound motions are straight when each of the components is always the same, although they may be unequal among themselves; or when the components are always equal among themselves, although they may not always be the same. Finally, it is visible that the lines described by these motions are curved when the components are unequal among themselves and are not always the same.

Finally, this line represents to the imagination all the places where this body, pushed by two different forces toward two different places, must be found; so that one can mark precisely the point where this body must be at any given instant. If, for example, one wishes to know where it must be at the beginning of the fourth minute, one need only divide the lines AB or AC into parts expressing the space that each of these known forces would be capable of making this body traverse in one minute; and take three of these parts in any of these lines, and then draw through the beginning of the fourth 3X parallel to AB, or IIIX parallel to AC. For it is evident that the point X, which one or the other of these parallels determines on the line AXYE, marks the place where this body will be at the beginning of the third minute of its motion. Thus this manner of examining questions not only sustains the mind’s vision, it even shows it the resolution; and it gives it enough light to discover unknown things from very few known things.

It suffices, for example, after what has been said, that one only knows that a body which was at A at such a time is found at E at another, and that different forces push it along lines making a given angle such as BAC, to discover the line of its compound motion, and the different degrees of velocity of the simple motions, provided one knows that these motions are equal among themselves or uniform. For when one has two points of a straight line, one has the whole; and one can compare the straight line AE, or the compound motion which is known, with the lines AB and AC, that is to say, with the simple motions which are unknown.

If one again supposes that a stone is pushed from A toward B by a uniform motion, but that it falls toward C, infinitely far from point A, by an unequal motion, similar to that by which heavy bodies are commonly thought to tend toward the center of the earth — that is to say, that the spaces it traverses are to each other as the squares of the times it takes to traverse them — the line it will describe will always be a parabola, and one will be able to determine with the utmost exactness the point where it will be at such a moment of its motion.

For if in this first moment this body falls two feet from A toward C, in the second six, in the third ten, in the fourth fourteen, and it is pushed by a uniform motion from A toward B of length sixteen feet, it is visible that the line it will describe will be a parabola whose parameter will be eight feet long. For the square of the ordinates to the diameter, which mark the times and the regular motion from A toward B, will be equal to the rectangle of the parameter by the lines which mark the unequal and accelerated motions; and the squares of the ordinates — that is to say, the squares of the times — will be to each other as the parts of the diameter between the pole and the ordinates. 16 : 64 :: 2 : 8. 64 : 144 :: 8 : 18, etc. It suffices to consider the sixth figure to be convinced of this. For the semicircles show that A2 is to A4 — that is, to the ordinate 2X, which is equal to it — as 2X is to A8; that A18 is to A12 — that is, to the ordinate 18X — as 18X is to A8, etc.; thus the rectangles A2 by A8, and A18 also by A8, are equal to the squares of 2X and 18X, etc., and consequently these squares are to each other as these rectangles. The parallels to AB and AC which intersect at points X, X, X further make sensibly known the path this body must follow. They mark the places where it must be at such a time. They finally represent to the eyes the true magnitude of the compound motion and of its acceleration at a determined time.

Again supposing that a body moves from A toward C unequally, as well as from A toward B, if the inequality is the same at the beginning and always — that is, if the inequality of its motion toward C is similar to that toward B, or if it increases in the same proportion — the line it will describe will be straight.

But if one supposes that there is inequality in the increase or decrease of the simple motions, although one may suppose this inequality to be whatever one wishes, it will always be easy to find the line representing to the imagination the compound motion of simple motions, by expressing these motions by lines and drawing parallels to these lines that intersect. For the line passing through all the intersections of these parallels will represent the compound motion of these unequal motions, unequally accelerated or diminished.

For example, if one supposes that a body is moved by two equal or unequal forces as one wishes, that one of these motions always increases or decreases according to any geometric or arithmetic progression one wishes, and that the other motion also increases or decreases according to any arithmetic or geometric progression one wishes; to find the points through which the line representing to the eyes and to the imagination the compound motion of these motions must pass, here is what must be done.

First, one must draw, as has been said, the two lines AB and AC, to express the two simple motions; and divide these lines according to the supposition of the acceleration of these motions. If one supposes that the motion expressed by the line AC increases or decreases according to an arithmetic progression 1, 2, 3, 4, 5; it must be divided at the points marked 1, 2, 3, 4, 5; and if one supposes that the motion expressed by the line AB increases according to the double progression 1, 2, 4, 8, 16, or decreases according to the sub-double progression 4, 2, 1, 1/2, 1/4, 1/8, it must be divided at the points marked 1, 2, 4, 8, 16; or 4, 2, 1, 1/2, 1/4, 1/8. Then one must draw through these divisions parallels to AB and AC; and the line AE, which must express the compound motion sought, will necessarily pass through all the points where these parallels intersect. And thus one sees the path this moving body must follow.

If one wishes to know exactly how long this body has been moving when it arrives at such a point, the parallels drawn from this point to AB or AC will mark it, for the divisions of AB and AC mark the time. Similarly, if one wishes to know the point this body will have reached at such a time, the parallels drawn from the divisions of the lines AB and AC representing this time will mark by their intersection this point sought. For the distance from the place where it began to move, it will always be easy to know by drawing a line from this point to A, for the length of this line will be known in relation to AB or AC which are known. But for the length of the path this body will have traversed to reach this point, it will be difficult to know, because the line of its motion AE being curved, it cannot be related to any of these straight lines.

But if one wished to determine the infinite points through which this body must pass — that is to say, to describe exactly and by a continuous motion the line AE — it would be necessary to have a compass whose movement of the legs was regulated according to the conditions expressed in the suppositions just made. This is often very difficult to invent, impossible to execute, and rather useless for discovering the relations things have among themselves, since one generally does not need all the points of which this line is composed, but only some which serve to guide the imagination when it considers such motions.

These examples suffice to show that one can express by lines and thus represent to the imagination most of our ideas; and that geometry, which teaches how to make all the comparisons necessary to know the relations of lines, is of much more extensive use than is ordinarily thought. For finally, astronomy, music, mechanics, and generally all the sciences that deal with things capable of more or less — and therefore can be regarded as extended — that is, all exact sciences, can be related to geometry, because all speculative truths consisting only in the relations of things and in the relations found between their relations, they can all be reduced to lines. One can geometrically draw several consequences from them; and these consequences, being rendered sensible by the lines representing them, it is almost impossible to make a mistake, and one can push the sciences very far with great ease.

The reason, for example, why one recognizes very distinctly and marks precisely in music an octave, a fifth, a fourth, is that one expresses sounds with exactly divided strings, and one knows that the string sounding the octave is in double proportion to the other with which the octave is made; that the fifth is in sesquialter proportion or three to two, and so on for the others. For the ear alone cannot judge sounds with the precision and exactness necessary for a science. The most skilled practitioners, those with the most delicate and fine ear, are not yet sensitive enough to recognize the difference between certain sounds; and they falsely persuade themselves that there is none, because they judge things only by the feeling they have of them. There are some who do not put any difference between an octave and three ditones. Some even imagine that the major tone is not different from the minor tone; so that the comma, which is the difference, is imperceptible to them, and a fortiori the schisma, which is only half of the comma.

It is therefore only reason that makes us clearly see that the space of the string, which makes the difference between certain sounds, being divisible into several parts, there may still be a very great number of different sounds, useful and useless for music, which the ear cannot discern. From which it is clear that without arithmetic and geometry regular and exact music would be unknown to us, and we could only succeed in this science by chance and by imagination; that is to say, music would no longer be a science founded on incontestable demonstrations, although the airs composed by the force of imagination are more beautiful and more agreeable to the senses than those composed by rules.

Similarly in mechanics, the weight of some weights and the distance of the center of gravity of this weight from the support being capable of more and less, both can be expressed by lines. Thus, one usefully employs geometry to discover and demonstrate an infinity of new inventions very useful to life, and even very agreeable to the mind because of the evidence accompanying them.

If, for example, one has a given weight, such as six pounds, which one wishes to balance with a weight of only three pounds, and this six-pound weight is attached to the arm of a balance at a distance of two feet from the support; knowing only the general principle of all mechanics — that weights to remain in equilibrium must be in reciprocal proportion to their distance from the support, that is to say, one weight must be to the other weight as the distance between the latter and the support is to the distance of the former from the same support — it will be easy to find, by geometry, what must be the distance of the three-pound weight so that everything remains in equilibrium, by finding, according to the twelfth proposition of the sixth book of Euclid, a fourth proportional line which will be four feet. So that, knowing only the fundamental principle of mechanics, one can discover with evidence all the truths depending on it by applying geometry to mechanics, that is, by sensibly expressing by lines all the things one considers in mechanics.

Lines and geometrical figures are therefore very suitable for representing to the imagination the relations between magnitudes or between things that differ by more and less, such as spaces, times, weights, etc., both because they are very simple objects and because one imagines them with great ease. One could even say, to the advantage of geometry, that lines can represent to the imagination more things than the mind can know, since lines can express the relations of incommensurable magnitudes — that is to say, magnitudes whose relations one cannot know, because they have no measure by which one can make comparison. But this advantage is not very considerable for the search for truth, since these sensible expressions of incommensurable magnitudes do not distinctly reveal to the mind their true magnitude.

Geometry is therefore very useful for making the mind attentive to the things whose relations one wishes to discover; but it must be admitted that it is sometimes an occasion of error for us, because we occupy ourselves so much with the evident and agreeable demonstrations that this science provides us, that we do not sufficiently consider nature. It is principally for this reason that all invented machines do not succeed, that all musical compositions or the proportions of consonances being best observed are not the most agreeable, and that the most exact computations in astronomy sometimes do not better predict the magnitude and time of eclipses. Nature is not abstract: the levers and wheels of mechanics are not lines and mathematical circles; our tastes for musical airs are not always the same in all men, nor in the same men at different times; they change according to the different emotions of the mind, so that there is nothing more bizarre. Finally, as regards astronomy, there is no perfect regularity in the course of the planets; swimming in these great spaces, they are irregularly carried by the fluid matter surrounding them. Thus, the errors one makes in astronomy, mechanics, music, and in all the sciences to which one applies geometry, do not come from geometry, which is an incontestable science, but from the false application one makes of it.

One supposes, for example, that the planets describe by their motions perfectly regular circles and ellipses; which is not true. One does well to suppose it, in order to reason, and also because it is not far from the truth; but one must always remember that the principle on which one reasons is a supposition. Similarly, in mechanics one supposes that wheels and levers are perfectly hard and similar to mathematical lines and circles without weight and without friction; or rather, one does not sufficiently consider their weight, their friction, their matter, nor the relation these things have among themselves; that hardness or size increases weight, that weight increases friction, that friction diminishes force, that it breaks or wears out the machine in a short time, and thus what almost always succeeds on a small scale almost never succeeds on a large scale.

One must not therefore be surprised if one is mistaken, since one wishes to reason on principles that are not exactly known; and one must not imagine that geometry is useless because it does not deliver us from all our errors. Suppositions established, it makes us reason consistently. Rendering us attentive to what we consider, it makes us know it evidently. We even recognize by it if our suppositions are false; for being always certain that our reasonings are true, and experience not agreeing with them, we discover that the supposed principles are false. But without geometry and arithmetic one can discover nothing in the exact sciences that is somewhat difficult, although one has certain and incontestable principles.

One must therefore regard geometry as a kind of universal science which opens the mind, renders it attentive, and gives it the skill to regulate its imagination and to draw from it all the aid it can receive: for by the aid of geometry, the mind regulates the movement of the imagination, and the regulated imagination sustains the vision and application of the mind. But so that one may know how to make good use of geometry, one must note that all things falling under the imagination cannot be imagined with equal facility; for all images do not equally fill the mind’s capacity. It is more difficult to imagine a solid than a plane, and a plane than a simple line: for there is more thought in the clear view of a solid than in the clear view of a plane and a line. It is the same with different lines; it takes more thought — that is to say, more mind capacity — to represent a parabolic or elliptical line, or some other more composed ones, than to represent the circumference of a circle, and more for the circumference of a circle than for a straight line, because it is more difficult to imagine lines described by highly composed motions and having several relations, than those described by very simple motions or having fewer relations. For relations, not being able to be clearly perceived without the mind’s attention to several things, require all the more thought to perceive them the more numerous they are. There are therefore figures so composed that the mind does not have enough capacity to imagine them distinctly; but there are also others that the mind imagines with great facility.

Of the three kinds of rectilinear angles — acute, right, and obtuse — only the right one awakens in the mind a distinct and well-terminated idea. There is an infinity of acute angles which all differ from each other; the same for obtuse ones. Thus, when one imagines an acute or obtuse angle, one imagines nothing exact nor distinct. But when one imagines a right angle, one cannot be mistaken: its idea is very distinct, and even the image one forms of it in the brain is ordinarily accurate enough.

It is true that one can also determine the vague idea of an acute angle to the particular idea of an angle of thirty degrees, and that the idea of an angle of thirty degrees is as exact as that of an angle of 90, that is, of a right angle. But the image one would try to form of it in the brain would not be, by a long way, as accurate as that of a right angle. One is not accustomed to representing this image, and one cannot trace it except by thinking of a circle or a determined part of a circle divided into equal parts. But to imagine a right angle, it is not necessary to think of this division of a circle; the sole idea of a perpendicular suffices for the imagination to trace the image of this angle, and one feels no difficulty in representing perpendiculars, because one is accustomed to seeing all things upright.

It is therefore easy to judge that to have a simple, distinct, well-terminated object, suitable to be imagined with facility, and consequently to render the mind attentive and preserve evidence in the truths it seeks, one must refer all the magnitudes we consider to simple surfaces terminated by lines and by right angles, such as perfect squares and other rectangular figures, or to simple straight lines; for these figures are those whose nature one knows most easily.

I could have attributed to the senses the aid one draws from geometry to preserve the mind’s attention; but I have believed that geometry belongs more to the imagination than to the senses, although lines are something sensible. It would be rather useless to deduce here the reasons I had, since they would only serve to justify the order I have kept in what I have just said, which is not essential. I have not spoken either of arithmetic or algebra, because the digits and letters of the alphabet used in these sciences are not so useful for increasing the mind’s attention as for increasing its capacity, as we will explain in the following chapter.

Such are the general aids that can render the mind more attentive. No others are known, except the will to have attention; of which one does not speak, because it is assumed that all those who study wish to be attentive to what they study.

There are nevertheless still several which are particular to certain persons, such as certain drinks, certain foods, certain places, certain dispositions of the body, and some other aids of which each must instruct himself by his own experience. One must observe the state of one’s imagination after meals and consider which things maintain or dissipate the attention of one’s mind. What can be said most generally is that moderate use of foods which produce many animal spirits is very suitable for increasing the attention of the mind and the force of the imagination in those who have it weak and languishing.