The Rules

Means of increasing the extent and capacity of the mind. That arithmetic and algebra are absolutely necessary for this.

One must not first imagine that one can ever truly increase the capacity and extent of one’s mind. The soul of man is, so to speak, a determinate quantity or a portion of thought which has limits it cannot surpass; the soul cannot become greater or more extensive than it is; it does not swell or expand as one believes liquids and metals do; in short, it seems to me that it never perceives more at one time than at another.

It is true that this seems contrary to experience. Often one thinks of many objects; often one thinks of only one, and often even one says that one thinks of nothing. However, if one considers that thought is to the soul what extension is to the body, one will clearly recognize that just as a body cannot truly be more extended at one time than at another, so, properly speaking, the soul can never think more at one time than at another, whether it perceives several objects, whether it perceives only one, or even during the time when one says one thinks of nothing.

But the reason why one imagines that one thinks more at one time than at another is that one does not distinguish enough between perceiving confusedly and perceiving distinctly. Undoubtedly, much more thought is needed, or the capacity one has to think must be more filled, in order to perceive several things distinctly than to perceive only one; but no more thought is needed to perceive several things confusedly than to perceive one thing distinctly. Thus, there is no more thought in the soul when it thinks of several objects than when it thinks of only one, since if it thinks of only one it always perceives much more clearly than when it applies itself to several.

For one must note that a perfectly simple perception sometimes contains as much thought, that is, it fills as much of the capacity the mind has for thinking, as a judgment and even as a composed reasoning, since experience teaches that a simple perception, but lively, clear and evident, of a single thing, applies and occupies us as much as a composed reasoning, or as the obscure and confused perception of several relations between several things.

For just as there is as much or more sensation in the close view of an object that I hold right near my eyes and examine carefully than in the view of an entire countryside that I look at negligently and without attention, so that the clarity of the sensation I have of the object close to my eyes compensates for the extent of the confused sensation I have of several things I see without attention in a countryside; so the mind’s view of a single object is sometimes so vivid and distinct that it contains as much or even more thought than the view of the relations that exist between several things.

It is true that at certain times it seems to us that we think of only one thing, and yet we have difficulty understanding it properly; and that at other times we understand that thing and several others with great ease. And from this we imagine that the soul has more extent or a greater capacity for thinking at one time than at another. But it is obvious that we are mistaken. The reason why, at certain times, we have difficulty conceiving the easiest things is not that the soul’s thought, or its capacity for thinking, is diminished; but it is that this capacity is filled by some vivid sensation of pain or pleasure, or by a great number of weak and obscure sensations, which produce a kind of dizziness; for dizziness is ordinarily only a confused sensation of a very great number of things.

A piece of wax is capable of one very distinct shape: it cannot receive two without one confusing the other, for it cannot be entirely round at the same time; finally, if it receives a million, none of them will be distinct. Now, if this piece of wax were capable of knowing its own shapes, it could not, however, know which shape would terminate it, if the number were too great. It is the same with our soul: when a very great number of modifications fill its capacity, it cannot perceive them distinctly, because it does not sense them separately. Thus, it thinks it senses nothing; it cannot say that it feels pain, pleasure, light, sound, tastes: it is none of those things, and yet it is only that which it senses.

But even if we were to suppose that the soul were not subject to the confused and disorderly movement of the animal spirits, and that it were so detached from its body that its thoughts did not depend on what takes place in it, it could still happen that we would understand certain things more easily at one time than at another, without the capacity of our soul diminishing or increasing; because then we would think of other things in particular, or of indeterminate and general being. I shall explain myself.

The general idea of the infinite is inseparable from the mind, and it entirely occupies its capacity when it does not think of something particular. For when we say we think of nothing, that does not mean we are not thinking of this idea, but simply that we are not thinking of something particular.

Certainly, if this idea did not fill our mind, we could not think of all sorts of things, as we can; for in short, one cannot think of things of which one has no knowledge. And if this idea were not more present to the mind when it seems to us we are thinking of nothing than when we are thinking of something particular, we would have as much ease in thinking of what we wanted when we are strongly applied to some particular truth as when we are applied to nothing, which is contrary to experience. For example, when we are strongly applied to some proposition of geometry, we do not have as much ease in thinking of all things as when we are occupied with no particular thought. Thus, one thinks more of general and infinite being when one thinks less of particular and finite beings; and one always thinks just as much at one time as at another.

One cannot, therefore, increase the extent and capacity of the mind by inflating it, so to speak, and giving it more reality than it has naturally, but only by managing it skillfully; which is done perfectly by arithmetic and algebra. For these sciences teach the means of abbreviating ideas in such a way and of considering them in such an order, that although the mind has little extent, it is capable, with the help of these sciences, of discovering very complex truths which at first appear incomprehensible. But one must take things from their principle in order to explain them with more solidity and clarity.

Truth is nothing other than a real relation, either of equality or of inequality. Falsity is only the negation of truth, or a false and imaginary relation. Truth is what is. Falsity is not; or, if one wishes, it is what is not. One is never mistaken when one sees the relations that exist, since one is never mistaken when one sees the truth. One is always mistaken when one judges that one sees certain relations and those relations do not exist; for then one sees falsity, one sees what is not, or rather one does not see, since nothingness is not visible and the false is a relation that does not exist. Whoever sees the relation of equality between two times two and four, sees a truth, because he sees a relation of equality which is such as he sees it. Likewise, whoever sees a relation of inequality between two times 2 and 5, sees a truth, because he sees a relation of inequality which exists. But whoever judges that he sees a relation of equality between two times 2 and 5, is mistaken, because he sees, or rather because he thinks he sees, a relation of equality which does not exist. Truths are therefore only relations, and the knowledge of truths is the knowledge of relations. But falsities are not, and the knowledge of falsity, or a false knowledge, is the knowledge of what is not, if that can be said; for just as one can know what is not only in relation to what is, one recognizes error only through truth.

One can distinguish as many kinds of falsities as truths. And just as there are relations of three sorts, from one idea to another idea, from a thing to its idea or from an idea to its thing, finally from one thing to another thing, there are truths and falsities of three sorts. There are some between ideas, between things and their ideas, and between things only. It is true that two times 2 are 4; it is false that two times 2 are 5: there is a truth and a falsity between ideas. It is true that there is a sun; it is false that there are two: there is a truth and a falsity between things and their ideas. It is true finally that the earth is larger than the moon, and it is false that the sun is smaller than the earth: there is a truth and a falsity that is only between things. Of these three sorts of truths, those that are between ideas are eternal and immutable, and, because of their immutability, they are also the rules and measures of all the others; for every rule or every measure must be invariable. And that is why one considers in arithmetic, algebra and geometry only these sorts of truths, because these general sciences regulate and contain all the particular sciences. All the relations or all the truths that are between created things, or between ideas and created things, are subject to the change of which every creature is capable. Only the truths that are between our ideas and sovereign being are immutable, like those that are between ideas alone, because God is not subject to change, any more than the ideas he contains.

It is also only the truths that are between ideas that one tries to discover by the sole exercise of the mind; for one almost always uses one’s senses to discover the other truths. One uses one’s eyes and hands to assure oneself of the existence of things, and to recognize the relations of equality or inequality that exist between them. It is only of ideas that the mind can infallibly know the relations by itself and without the use of the senses. But not only is there relation between ideas, but also between the relations that are between ideas, between the relations of relations of ideas, and finally between assemblages of several relations and between the relations of these assemblages of relations, and so on to infinity; that is to say, there are truths composed to infinity. In geometrical terms, one calls a simple truth, that is, the relation of one whole idea to another, the relation of 4 to 2, or of two times 2, a geometric ratio, or simply a ratio; for the excess or defect of one idea over another, or, to use ordinary terms, the excess or defect of a magnitude is not properly a ratio; nor are equal excesses or defects of magnitudes equal ratios. When the ideas or magnitudes are equal, it is a ratio of equality; when they are unequal, the ratio is one of inequality.

The relation that is between the relations of magnitudes, that is, between ratios, is called a compounded ratio, because it is a compounded relation; the relation that is the relation of 6 to 4 and of 3 to 2 is a compounded ratio. And when the component ratios are equal, this compounded ratio is called a proportion or duplicate ratio. The relation that is between the ratio of 8 to 4 and the ratio of 6 to 3 is a proportion, because these two ratios are equal.

Now, one must note that all relations or all ratios, both simple and compounded, are true magnitudes; and that the very term magnitude is a relative term which necessarily indicates some relation; for nothing is great by itself and without relation to something else, except the infinite or unity. All whole numbers are even relations as truly as fractional numbers, or numbers compared to another, or divided by another, although one may not reflect upon it, because these whole numbers can be expressed by a single digit, 4, for example, but 8/2 is a relation as true as 1/4 or 2/8. The unity to which it has relation is not expressed, but it is understood; for 4 is a relation just as much as 4/1 or 8/2, since 4 is equal to 4/1 or to 8/2. Every magnitude being therefore a relation, or every relation a magnitude, it is evident that one can express all relations by numbers, and represent them to the imagination by lines.

Thus, all truths being only relations; to know exactly all truths, both simple and compounded, it suffices to know exactly all relations, both simple and compounded. There are two sorts of them, as has just been said: relations of equality and inequality. It is evident that all relations of equality are similar, and that as soon as one knows that one thing is equal to another known thing, one knows exactly its relation. But it is not the same with inequality: one knows that a tower is larger than a fathom and smaller than a thousand fathoms, and yet one does not know exactly its size and the relation it has to a fathom.

To compare things with each other, or rather to measure exactly the relations of inequality, one needs an exact measure, one needs a simple and perfectly intelligible idea, a universal measure that can accommodate all sorts of subjects. This measure is unity; it is by it that one measures all things exactly, and without it it is impossible to know anything with any exactitude. But since all numbers are composed only of unity, it is already evident that, without the ideas of numbers, and without the manner of comparing and measuring these ideas, that is, without arithmetic, it is impossible to advance in the knowledge of compounded truths.

The ideas or the relations between ideas, in a word the magnitudes, being able to be larger and smaller than other magnitudes, one can make them equal only by more and by less, together with unity repeated as many times as necessary. Thus, it is only by the addition and subtraction of unity and the parts of unity (when one conceives it divided) that one measures exactly all magnitudes and discovers all truths. Now, of all the sciences, arithmetic and above all algebra are the only ones that teach us to perform these operations with skill, with clarity, and with an admirable economizing of the mind’s capacity. These two sciences are therefore the only ones that give to the mind all the perfection and all the extent of which it is capable, since it is by them alone that one discovers all the truths that can be known with complete exactitude.

Ordinary geometry does not perfect the mind so much as the imagination, and the truths one discovers by this science are not always as evident as geometers imagine. They think, for example, that they have expressed the value of certain magnitudes when they have proved that they are equal to certain lines, which are the subtenses of right angles whose sides are exactly known, or to others that are determined by some conic section. But it is evident that they are mistaken; for these subtenses, for example, are themselves unknown. One knows √8 or √20 more exactly than a line that one imagines or that one draws on paper, to serve as the subtense of a right angle whose sides are 2, or whose one side is 2 and the other 4. One knows at least that √8 is close to 3, and that √20 is about 4 and 1/2; and one can, by certain rules, always approach infinitely close to their true magnitude; and if one cannot reach it, it is because the mind cannot comprehend the infinite. But one has only a very confused idea of the magnitude of the subtenses, and one is even obliged to have recourse to √8 or √20 to express them. Thus, the geometrical constructions that one uses to express the values of unknown quantities are not so useful for regulating the mind and discovering the relations or truths one seeks as for regulating the imagination. But since one takes much more pleasure in using one’s imagination than one’s mind, mathematicians ordinarily have more esteem for geometry than for arithmetic and algebra.

To make it perfectly understood that arithmetic and algebra together are the true logic that serves to discover truth, and to give the mind all the extent of which it is capable, it suffices to make some reflections on the rules of these sciences. It has just been said that all truths are only relations; that the simplest and best known of all relations is that of equality; that it is the starting point from which one must measure the others to have an idea of inequality; that the measure one is obliged to use is unity; and that one must add or subtract it as many times as necessary to measure the excess or defect of the inequality of these magnitudes;

From this it is clear that all the operations that can serve to discover relations of equality are only additions and subtractions: additions of magnitudes to equalize magnitudes, additions of ratios to equalize ratios, or to put magnitudes in proportion; finally addition of ratios of ratios to equalize ratios of ratios, or to put magnitudes in compounded proportion.

To equalize 4 with 2, one has only to add 2 to 2, or subtract 2 from 4, or finally add unity to 2 and subtract it from 4. This is clear.

To equalize the ratio of 8 to 2 with the ratio of 6 to 3, one must not add 3 to 2 or subtract 3 from 8, so that the excess of one number over the other is equal to 3, which is the excess of 6 over 3; that would only be adding and equalizing simple magnitudes, the excess of 8 over 5 to that of 6 over 3. One must first seek the magnitude of the ratio of 8 to 2, or what 8/2 is worth, and one finds by dividing 8 by 2 that the exponent of this ratio is 4, or that 8/2 is equal to 4. One must likewise see what is the magnitude of the ratio of 6 to 3, and one finds it is equal to 2.

Thus one recognizes that these two ratios 8/2 equal to 4, and 6/3 equal to 2, differ only by 2. So that to equalize them one can either add to 6/3 another 6/3 equal to 2, for one will have 12/3, which will be a ratio equal to 8/2, or subtract 4/2 equal to 2 from 8/2, for one will have 4/1 which will be a ratio equal to 6/3; or finally add unity to 6/3 and subtract it from 2, for one will have 9/3 and 6/2, which are equal ratios, for 9 is to 3 as 6 is to 2.

To find the magnitude of the inequality between the ratios that result, one from the compounded ratio of 12 to 3 and of 3 to 1, and the other from the compounded ratio of 8 to 2 and of 2 to 1, one must follow the same path.

First, the magnitude of the ratio of 12 to 3 is denoted by 4, where 4 is the exponent of the ratio of 12 to 3, and 3 is the exponent of 3 to 1, and the exponent of the ratio of exponents 4 and 3 is 4/3. Second, the exponent of 8 to 2 is 4, and of 2 to 1 is 2, and the exponent of exponents 4 and 2 is 2; finally the inequality between the ratios that result from the ratios of ratios is the difference between 4/3 and 2, that is, 1/3.

Therefore 1/3 added to the ratio of the ratios 12 to 3 and 3 to 1, or subtracted from the ratio of the other ratios 8 to 2 and 2 to 1, equalizes these ratios of ratios, and produces a compounded proportion. Thus one can use additions and subtractions to equalize magnitudes and their ratios, both simple and compounded, and to have an exact idea of the magnitude of their inequality.

One uses multiplications and divisions, both simple and compounded, but these are only compounded additions and subtractions. Multiplying 4 by 3 is doing as many additions of 4 as 3 contains additions of unity, or finding a number that has the same ratio to 4 as 3 has to unity; and dividing 12 by 4 is subtracting 4 from 12 as many times as possible, that is, finding a ratio to unity equal to that of 12 to 4; for 3, which will be its exponent, has the same ratio to unity as 12 to 4. The extraction of square roots, cube roots, etc., are only divisions by which one seeks one, two or three mean proportionals.

Man’s:

• mind is so small
• memory is so unfaithful
• imagination is so limited

Without the use of numbers and writing, and without the skill used in arithmetic, it would be impossible to perform the operations necessary to know the inequality of magnitudes and their ratios.

When there would be several numbers to add or subtract, or, what is the same thing, when these numbers are large, and one can add them only by parts, one would always forget some. There is no imagination extensive enough to add together somewhat large fractions, like 1703/4093, 17946103/10431, or to subtract one from the other. Multiplications, divisions and extractions of roots of whole numbers are infinitely more embarrassing than simple additions or subtractions; the mind alone, without the help of arithmetic, is too small and too weak to perform them, and it is useless for me to stop here to show this.

However, analysis or algebra is different from arithmetic.

It much less divides the capacity of the mind, it abbreviates ideas in the simplest and easiest way that can be conceived. What is done in much time by arithmetic is done in a moment by algebra, without the mind becoming confused by the change of numbers and the length of the operations.

A particular operation of arithmetic discovers only one truth; a similar operation of algebra discovers an infinity. Finally, there were things that could be known, and that it was necessary to know, of which one could not have knowledge by the use of arithmetic alone; but I do not believe there is anything useful, and that men can know with exactitude, of which they cannot have knowledge by arithmetic and by algebra. So that these two sciences are the foundation of all the others, and give the true means of acquiring all the exact sciences, because one cannot economize the mind’s capacity more than is done by arithmetic, and above all by algebra.