Application of the Other Rules to Particular Questions

There are 2 sorts of questions:

1. Simple
2. Compound

The resolution of the former depends solely on the mind’s attention to the clear ideas of the terms that express them.

The latter can only be resolved by comparison to a third idea or to several other ideas; one cannot discover the unknown relations expressed by the terms of the question by immediately comparing the ideas of those terms, for they cannot be joined or compared.

One therefore requires one or more intermediate ideas in order to make the necessary comparisons to discover these relations, and one must observe exactly that these intermediate ideas be clear and distinct, in proportion as one endeavors to discover more exact relations and a greater number of them.

This rule is merely a consequence of the first, and it is of equal importance. For if it is necessary, in order to know exactly the relations of the things one compares, to have clear and distinct ideas of them, it is necessary, for the same reason, to know well the intermediate ideas by which one intends to make these comparisons, since one must know distinctly the relation of the measure to each of the things one measures in order to discover their relations. Here are some examples.

When a very light little vessel containing a magnet is left to float freely, and one presents to the northern pole of this magnet the same pole of another magnet held in one’s hands, one immediately sees the first magnet withdraw as if pushed by some violent wind. And one desires to know the cause of this effect.

It is quite visible that, to account for the movement of this magnet, it is not enough to know the relations it has with the other; for even if one knew all of them perfectly, one could not comprehend how these two bodies could push each other without meeting.

One must therefore examine what things one knows distinctly to be capable, according to the order of nature, of moving some body; for the question is to discover the natural cause of the magnet’s movement, which is certainly a body.

Thus, one must not resort to some quality, some form, or some entity that one does not clearly know to be capable of moving bodies, nor even to some intelligence, for one does not know with certainty that intelligences are the ordinary causes of the natural movements of bodies, nor even if they can produce movement.

One knows evidently that it is a law of nature that bodies move one another when they meet; one must therefore endeavor to explain the movement of the magnet by means of some body that meets it. It is true that there might be something other than a body that moves it; but if one has no distinct idea of this thing, one must not use it as an acceptable means to discover what one seeks, nor to explain it to others; for it is not accounting for an effect to give as its cause a thing that no one clearly conceives. One must therefore not trouble oneself whether there is or is not some other natural cause of the movement of bodies besides their mutual encounter; one must rather suppose that there is none, and consider with attention what body can meet and move this magnet.

One sees at first that it is not the magnet held in the hand, since it does not touch the one that is moved. But because the floating magnet is moved only upon the approach of the one held in the hand, and does not move itself, one must conclude that, although it is not the magnet held in the hand that moves it, it must be some small bodies that issue from it and are pushed by it toward the other magnet.

To discover these small bodies, one must not simply open one’s eyes and draw near to the magnet, for the senses would mislead reason, and one might perhaps judge that nothing issues from the magnet, because one sees nothing issuing from it; or one might forget that one does not see the winds, even the most impetuous, nor many other bodies that produce extraordinary effects. One must hold firm to this very clear and very intelligible means, and examine carefully all the effects of the magnet, in order to discover how it can ceaselessly push out these small bodies without diminishing; for the experiments one makes will reveal that these small bodies, which issue from one side, immediately re-enter by the other, and they will serve to explain all the difficulties that can be raised against the manner of resolving this question. But one must carefully note that one should not abandon this means, even if one cannot answer some difficulties based on the ignorance in which one is of many things.

If one does not wish to examine why magnets repel each other when their same poles are opposed, but rather why they approach and join together when the northern pole of one is presented to the southern pole of the other, the question will be more difficult and a single means will not suffice to resolve it. It is not enough to know exactly the relations between the poles of these two magnets, nor to resort to the means taken for the previous question, for this means seems rather to prevent the effect whose cause is being sought. One must not resort either to any of the things we do not clearly know to be the natural and ordinary causes of bodily movements, nor relieve oneself of the difficulty of the question by the vague and indeterminate idea of an occult quality in the magnets, by which they attract one another; for the mind cannot clearly conceive that one body can attract another.

The impenetrability of bodies makes it clearly conceivable that motion can be communicated by impulse, and experience proves, without any obscurity, that it is indeed communicated by this means. But there is no reason nor any experience that clearly demonstrates the movement of attraction; for in the experiments that seem most suited to prove this sort of movement, one visibly recognizes, when one discovers its true and certain cause, that what appeared to be done by attraction is done only by impulse. Thus, one must not dwell on any other communication of motion than that which is done by impulse, since this manner is certain and incontestable, and there is at least some obscurity in the others one might imagine. But even if one could demonstrate that there are in purely corporeal things other principles of movement besides the encounter of bodies, one could not reasonably reject this one; one ought even to dwell on it preferably to any other, since it is the clearest and most evident, and it appears so incontestable that one does not hesitate to assert that it has been received by all peoples and in all times.

Experience shows that a magnet floating freely on water approaches the one held in the hand when a certain side is presented to it; one must therefore conclude that it is pushed toward it. But since it is not the magnet held in the hand that pushes the floating one, because the floating one approaches the one held, and yet the floating one would not move if the one held were not presented to it, it is evident that one must resort to at least two means to explain this question, if one wishes to resolve it by the received principle of the communication of movements.

Magnet c approaches magnet C; therefore the air, or the fluid and invisible matter surrounding it, pushes it, since there is no other body that can push it, and this is the first means.

Magnet c approaches only in the presence of magnet C; therefore it is necessary that magnet C determines the air to push magnet c, and this is the second means. It is evident that these two means are absolutely necessary. So the difficulty is now reduced to joining these two means together, which can be done in two ways: either by starting with something known in the air surrounding magnet c, or by starting with something known in magnet C.

If one knows that the parts of the air and of all fluid bodies are in continual agitation, one cannot doubt that they ceaselessly strike against magnet c, which they surround; and because they strike it equally on all sides, they do not push it more on one side than on the other, as long as there is as much air on one side as on the other. Things being thus, it is easy to judge that magnet C prevents there being as much of this air of which we speak toward a as toward b. But this can only be done by spreading some other bodies in the space between C and c: small bodies must therefore issue from the magnets to occupy this space. And this is also what experience shows when iron filings are scattered around a magnet; for these filings render visible the course of these invisible small bodies. Thus these small bodies, driving out the air that is toward a, magnet c is less pushed from that side than from the other; and consequently it must approach magnet C, since every body must move toward the side from which it is least pushed.

But if magnet c did not have, toward pole a, several pores suited to receive the small bodies issuing from pole B of the other magnet, and too small to receive those of the air, both gross and subtle, it is evident that these small bodies, being more agitated than this air, since they must drive it out from between the magnets, would push magnet c and drive it away from C. Thus, since magnet c approaches or recedes from C when different poles are presented to it, it is necessary to conclude that the small bodies issuing from magnet C pass freely and without repelling magnet c on side n, and repel it on side b. What I say of one of these magnets must also be understood of the other.

It is visible that one always learns something by this manner of reasoning on clear ideas and incontestable principles. For one has discovered that the air surrounding magnet c is driven out from between the magnets by bodies that ceaselessly issue from their poles, and which find their passage free on one side and closed on the other. And if one wished to discover what is roughly the size and shape of the pores of the magnet through which these small bodies pass, one would have to make other experiments; but that would lead us where we do not wish to go, and where we might well lose our way. One can consult on these questions the principles of the philosophy of M. Descartes, not to follow blindly the sentiments of this learned philosopher, but to accustom oneself to his method of philosophizing. I shall say only, to answer an objection that strikes at first glance, why these small bodies cannot re-enter by the pores from which they issued: that besides a determinate size or shape capable of producing this effect, the inflection of the little branches that compose these pores can yield in one direction to the small bodies that pass through them, and bristle up and close the passage to them in the other direction. The continual current of subtle matter from one pole to the other, within the pores of the magnet, is even sufficient to prevent it from re-entering by the pores from which it issued; for a part of this matter cannot overcome this current to force a passage through the pores from which it issued, nor through those of the same-named pole, which have a contrary current. So one must not be too surprised at the difference of the poles of the magnet; for this difference can be explained in many ways, and the only difficulty lies in recognizing the true one.

If one had tried to resolve the question just examined by starting with the small bodies supposed to issue from magnet C, one would have found the same thing, and one would also have discovered that the air is composed of an infinity of parts that are in continual agitation; for without that it would be impossible for magnet c to approach magnet C. I do not dwell on explaining this, because it is not difficult.

Here is a question more compound than the preceding ones, and one in which several rules must be employed. The question is: what might be the natural and mechanical cause of the movement of our limbs.

The idea of a natural cause is clear and distinct, if understood as I explained in the previous question; but the term “movement of our limbs” is equivocal and confused, for there are several sorts of these movements: there are voluntary, natural, and convulsive ones. There are also different limbs in the human body. Thus, according to the first rule, I must ask which of these movements one wishes to know the cause of. But if the question is left undetermined, so that I may use my own choice, I examine the question in this way.

I consider with attention the properties of these movements: and because I first discover that voluntary movements are ordinarily made more promptly than convulsive ones, I conclude that their cause may be different. Thus I can and consequently must examine the question in parts; for it appears to be of long discussion.

I restrict myself at first to considering only voluntary movement; and since we have several parts that serve these movements, I focus solely on the arm. I consider therefore that the arm is composed of several muscles which have almost all some action when one lifts something from the ground or moves some body in various ways; but I dwell only on one, being willing to suppose that the others are formed in roughly the same manner. I instruct myself in its composition by some book of anatomy or rather by the sensible view of its fibers and tendons, which I have dissected by some skillful anatomist to whom I put all the questions that may later give rise in my mind to some means of finding what I seek.

Considering therefore all things with attention, I cannot doubt that the principle of the movement of my arm depends on the shortening of the muscles that compose it. And if I am willing, to avoid embarrassing myself with too many things, to suppose, according to common opinion, that this shortening is done by means of the animal spirits that fill the belly of these muscles and thus bring their extremities closer, the whole question regarding voluntary movement will be reduced to knowing how the few animal spirits contained in an arm can suddenly inflate its muscles according to the orders of the will with a force sufficient to lift a burden of a hundredweight and more.

When one meditates upon this with some application, the first means that presents itself to the imagination is usually that of some sudden and violent effervescence similar to that of gunpowder or of certain liquors filled with alkaline salts, when they are mixed with those that are strong or full of acid salt. A little gunpowder is capable, when ignited, of lifting not only a burden of a hundredweight, but a tower and even a mountain. Earthquakes that overturn cities and shake entire provinces are also caused by spirits that ignite underground much like gunpowder. Thus, supposing in the arm a cause of the fermentation and dilation of the spirits, one could say that it is the principle of that force which men have to make such prompt and violent movements.

However, since one must distrust those means that enter the mind only through the senses and of which one has no clear and evident knowledge, one must not so readily use this one. For in the end, it is not enough to account for the force and promptness of our movements by a comparison. This reason is confused, but moreover it is imperfect; for one must explain here a voluntary movement, and fermentation is not voluntary. The blood ferments excessively in fevers, and one cannot prevent it. The spirits inflame and agitate in the brain, and their agitation does not diminish according to our desires. When a man moves his arm in various ways, it would be necessary, according to this explanation, that a million fermentations, large and small, prompt and slow, should begin and, what is even more difficult to explain according to this supposition, end at the very moment he wills it. It would be necessary that these fermentations did not dissipate all their matter and that this matter was always ready to catch fire. When a man has walked ten leagues, how many thousands of times must the muscles that serve for walking have filled and emptied? And how many spirits would be needed if fermentation dissipated and extinguished them at every step? This reason is therefore imperfect for explaining the movements of our body which depend entirely on our will.

It is evident that the present question consists in this problem of mechanics: To find, by means of pneumatic machines, a way to overcome a given force, say of a hundredweight, by another force as small as one wishes, such as the weight of an ounce, and that the application of this small force to produce its effect depends on the will. Now, this problem is easy to resolve and the demonstration of it is clear. One can resolve it by a vessel that has two openings, one of which is a little more than 1600 times larger than the other, and into which are inserted the nozzles of two equal bellows, and that one applies a force 1600 times only greater than the other to the bellows of the larger opening, for then the force 1600 times smaller will overcome the larger. And the demonstration of this is clear by mechanics, since the forces are not exactly in proportion to the openings, and the ratio of the small force to the small opening is greater than the ratio of the large force to the large opening.

But to resolve this problem by a machine that better represents the effect of the muscles than the one just given, one must blow a little into a balloon and then press on this balloon, half-inflated with wind, a stone of 5 or 6 hundredweight, or, having placed it on a table, cover it with a board, and this board with a very heavy stone, or make one of the heaviest men sit on this board, giving him even the freedom to hold onto something in order to resist the inflation of the balloon; for if someone blows again merely with his mouth into this balloon, he will lift the stone that compresses it or the man sitting on it, provided the channel by which the wind enters the balloon has a valve that prevents it from escaping when one needs to catch one’s breath. The reason for this is that the opening of the balloon is so small or must be supposed so small in relation to the entire capacity of the same balloon that resists by the weight of the stone, that a very small force is capable of overcoming a very large one by this means.

If one also considers that the breath alone is capable of pushing a lead ball violently by means of blowpipes, because the force of the breath is not dissipated and is constantly renewed, one will visibly recognize that, supposing the necessary proportion between the opening and the capacity of the balloon, the breath alone can easily overcome very great forces.

If therefore one conceives that the whole muscles or each of the fibers that compose them have, like this balloon, a capacity suited to receive the animal spirits; that the pores by which the spirits insinuate themselves into them are perhaps even smaller in proportion than the neck of a bladder or the hole of a balloon; that the spirits are retained and pushed in the nerves much like the breath in the blowpipes; and that the spirits are more agitated than the air in the lungs and pushed with more force into the muscles than it is into the balloons; one will recognize that the movement of the spirits that spread into the muscles can overcome the force of the heaviest burdens one carries; and that if one cannot carry heavier ones, the defect of force comes not so much from the side of the spirits as from that of the fibers and skins that compose the muscles, which would burst if too much effort were made. Moreover, if one takes heed that by the laws of the union of the soul and the body the movements of these spirits, as to their determination, depend on the will of men, one will see well that the movements of the arms must be voluntary.

It is true that we move our arm with such promptness that it seems at first incredible that the pouring of the spirits into the muscles that compose it could be prompt enough for that. But we must consider that these spirits are extremely agitated, always ready to pass from one muscle into another, and that not many are needed to inflate them as little as is necessary to move them alone, or when we lift something very light from the ground; for when we have something heavy to lift, we cannot do it with much promptness. The burdens being heavy, the muscles must be greatly inflated and tensed; to inflate them in this way, more spirits are needed than there are in the neighboring or antagonistic muscles. It therefore takes a little time to bring these spirits from afar and to push a quantity capable of resisting the weight. Thus those who are laden cannot run, and those who lift something heavy from the ground do not do it with as much promptness as those who lift a straw.

If one further reflects that those who have more fire or a little wine in their heads are much more prompt than others: that among animals those that have more agitated spirits, like birds, move with more promptness than those that have cold blood, like frogs, and that there are even some, like the chameleon, the tortoise, and some insects, whose spirits are so little agitated that their muscles do not fill any more promptly than a little balloon into which one would blow; if one considers all these things well, one may perhaps believe that the explanation we have just given is acceptable.

But although this part of the proposed question regarding voluntary movements is sufficiently resolved, one must not however assert that it is entirely resolved and that there is nothing more in our body that contributes to these movements than what has been said; for apparently there are in our muscles a thousand springs that facilitate these movements, which will be eternally unknown even to those who guess best about the works of God.

The second part of the question to be examined concerns natural movements, or those sorts of movements that have nothing extraordinary about them, as convulsive movements do, but which are absolutely necessary for the preservation of the machine, and which, consequently, do not depend entirely on our wills.

I consider therefore first with all the attention of which I am capable what are the movements that have these conditions, and if they are entirely similar. But because I immediately recognize that they are almost all different from one another; to avoid embarrassing myself with too many things, I dwell only on the movement of the heart. This part is the most known, and its movements are the most sensible. I examine therefore its structure, and I remark two things among many others: the first, that it is composed of fibers like the other muscles; the second, that it has two very considerable cavities. I judge therefore that its movement can be done by means of the animal spirits, since it is a muscle; and that the blood ferments and dilates in it, since there are cavities. The first of these judgments is based on what I have just said, and the second on the fact that the heart is much hotter than all the other parts of the body; that it is the one that spreads heat with the blood into all our limbs; that these two cavities could not have been preserved except by the dilation of the blood, and that thus they serve the cause that produced them. I can therefore account sufficiently for the movement of the heart by the spirits that agitate it and by the blood that dilates it when this blood ferments; for even though the cause I bring for its movement is perhaps not the true one, it seems to me certain that it is sufficient to produce it.

It is true that the principle of the fermentation or dilation of liquors is perhaps not known enough to all those who will read this to claim to have explained an effect when one has shown in general that its cause is fermentation; but one must not resolve all particular questions by going back to the first causes. It is not that one cannot go back to them and thus discover the true system on which all particular effects depend, provided one dwells only on clear ideas; but it is that this manner of philosophizing is not the most exact nor the shortest.

To make clear what I mean, one must know that there are two sorts of questions. In the first, the goal is to discover the nature and properties of something; in the others, one only wishes to know if such a thing has or does not have such a property: or, if one knows it has such a property, one only wishes to discover what its cause is. To resolve questions of the first kind, one must consider things in their birth, and always conceive them as being generated by the simplest and most natural ways. To resolve the others, one must go about it in a very different manner: one must resolve them by suppositions, and examine if these suppositions lead to some absurdity or if they lead to some clearly known truth.

One wishes, for example, to discover what are the properties of the roulette or of some one of the conic sections; one must consider these lines in their generation, and form them according to the simplest and least encumbered ways; for that is the best and shortest path to discover their nature and properties. One sees without pain that the sub-tense of the roulette is equal to the circle that formed it; and if one does not easily discover many of its properties by this way, it is because the circular line that serves to form it is not known enough. But for purely mathematical lines, or those whose relations can be more clearly known, such as the conic sections, it suffices, to discover a very great number of their properties, to consider these lines in their generation. One must only take care that, being able to generate them by regulated movements in several ways, not every sort of generation is equally suited to enlighten the mind; that the simplest are the best; and that it happens, however, that certain particular ways are more suited than others to demonstrate some particular properties.

But if the question is not to discover in general the properties of a thing, but to know if a thing has such a property, then one must suppose that it actually has it, and examine with attention what must follow from this supposition, if it leads to a manifest absurdity or to some incontestable truth that can serve as a means to discover what one seeks; and this is the manner in which geometers use to resolve their problems. They suppose what they seek and they examine what must happen from it; they consider attentively the relations that result from their supposition; they represent all these relations, which contain the conditions of the problem, by equations, and they then reduce these equations according to the rules they have for them, so that what is unknown is found equal to one or several things entirely known.

If therefore the question is to discover in general the nature of fire and the different fermentations that are the most universal causes of natural effects, I say that the shortest and surest way is to examine it in its principle. One must consider the formation of the most agitated bodies and whose movement spreads into those that ferment; one must, by clear ideas and by the simplest ways, examine what movement is capable of producing in matter; and because fire and the different fermentations are very general things, and which consequently depend on few causes, it will not be necessary to consider for a long time what matter is capable of, when it is animated by movement, to recognize the nature of fermentation in its principle; and one will learn at the same time many other things absolutely necessary for the knowledge of physics. Whereas, if one wished to reason in this question by suppositions, in order to go back thus to the first causes and to the laws of nature according to which all things are formed, one would make many false suppositions that would serve for nothing.

One might well recognize that the cause of fermentation is the movement of an invisible matter that is communicated to the parts of that which is agitated; for one knows well enough that fire and the different fermentations of bodies consist in their agitation, and that, by the laws of nature, bodies receive their movement immediately only by the encounter of some others more agitated. Thus one could discover that there is an invisible matter whose agitation is communicated by fermentation to visible bodies. But it would be morally impossible, by the way of suppositions, to discover how this is done; and it is not nearly so difficult to discover it when one examines the formation of the elements or bodies, of which there is a greater number of the same nature, as one can see in part by the system of M. Descartes.

The third part of the question, which is of convulsive movements, will not be extremely difficult to resolve, provided one supposes that there are in the body animal spirits capable of some fermentation and humors penetrating enough to insinuate themselves into the pores of the nerves by which the spirits spread into the muscles, provided also that one does not pretend to determine what is the true disposition of the invisible parts that contribute to these convulsive movements.

When one has separated a muscle from the rest of the body, and holds it by the extremities, one sees sensibly that it makes an effort to shorten itself when it is pricked in the belly. There is appearance that this depends on the construction of the imperceptible parts that compose it, which, like so many springs, are determined to certain movements by that of the prick. But who could assure himself of having found the true disposition of the parts that serve to produce this movement, and who could give an incontestable demonstration of it? Certainly this appears impossible, although perhaps, by thinking about it a lot, one could imagine a construction of muscles suited to make all the movements we see them capable of. One must therefore not think of determining what is the true construction of the muscles. But because one cannot reasonably doubt that there are spirits susceptible of some fermentation by the mixture of some subtle matter, and that acrid and pungent humors can insinuate themselves into the nerves, one can suppose it.

To resolve the proposed question, one must therefore examine first how many sorts of convulsive movements there are; and because the number of them appears indefinite, one must dwell on the principal ones, whose causes seem to be different. One must consider the parts in which they are made, the diseases that precede and follow them; if they are made with pain or without pain; and above all, what is their promptness and violence: for there are some that are made with promptness and violence, others with promptness without violence, and others with violence without promptness, and others finally without violence and without promptness; there are some that end and begin ceaselessly; there are some that hold the parts stiff and without movement for some time, and there are some that take away entirely the use of them and disfigure them.

All these things considered, it is not difficult to explain in general how these convulsive movements can be made after what has just been said of natural movements and voluntary movements; for if one conceives that there is mixed with the spirits contained in a muscle some matter capable of fermenting them, this muscle will inflate and produce in this part a convulsive movement.

If one can easily resist this movement, it will be a sign that the nerves will not be blocked by some humor, since one can empty the muscle of the spirits that have entered it and determine them to inflate the antagonistic muscle; but if one cannot, one must conclude that the pungent and penetrating humors have at least some part in this movement. It may even sometimes happen that these humors are the sole cause of these convulsive movements; for they can determine the course of the spirits toward certain muscles, by opening the passages that carry them there and closing the others, besides that they can shorten their tendons and fibers by penetrating their pores.

When a very heavy weight hangs at the end of a rope, one raises it notably if one merely wets this rope, because the parts of the water, insinuating themselves like so many little wedges between the threads of which the rope is composed, shorten it by widening it. Likewise the penetrating and pungent humors insinuating themselves into the pores of the nerves, shorten them, pull the parts attached to them, and produce in the body convulsive movements that are extremely slow, violent, and painful, and often leave the part in an extraordinary contortion for a considerable time.

As for the convulsive movements that are made with promptness, they are caused by the spirits; but it is not necessary that these spirits receive some fermentation, it suffices for that that the conduits by which they pass are more open on one side than on the other.

When all the parts of the body are in their natural situation, the animal spirits spread equally and promptly into them according to the need of the machine, and they faithfully execute the orders of the will; but when the humors trouble the disposition of the brain, and change or move diversely the openings of the nerves, or when, penetrating into the muscles, they agitate their springs, the spirits spread into the parts in a completely new manner, and produce extraordinary movements without the will having any part in it.

However, one can sometimes, by a strong resistance, prevent some of these movements, and even diminish little by little the traces that serve to produce them, although the habit is entirely formed. Those who pay attention to themselves prevent themselves easily enough from making grimaces or taking an indecent air or posture, although the body is disposed to it; they even overcome these things, although they are fortified by habit, but with much more pain, for one must always combat them in their birth and before the course of the spirits has made a path too difficult to close.

The cause of these movements is sometimes in the muscle that is agitated; it is some humor that pricks it or some spirits that ferment in it; but one must judge that it is in the brain, principally when the convulsions agitate not only one or two parts of the body in particular, but almost all of them, and also in several diseases that change the natural constitution of the blood and the spirits.

It is true that a single nerve having sometimes different branches that spread into parts of the body quite distant, as on the face and in the entrails, it happens often enough that the convulsion, having its cause in a part into which some of these branches insinuate, can communicate itself to those where the other branches respond without the brain being the cause and the spirits being corrupted. But when the convulsive movements are common to almost all the parts of the body, it is necessary to say either that the spirits ferment in an extraordinary manner, or that the order and arrangement of the parts of the brain is troubled, or that both these things happen. I do not dwell any longer on this question, for it becomes so compound and depends on so many things, when one descends into particulars, that it cannot easily serve to explain clearly the rules that have been given.

There is no science that provides more examples proper to show the utility of these rules than geometry, and principally algebra; for these two sciences make a continual use of them. Geometry makes clearly known the necessity of always beginning with the simplest things and those that contain the fewest relations. It always examines these relations by clearly known measures; it retrenches all that is useless to discover them; it divides compound questions into parts; it arranges these parts and examines them in order; finally, the only defect encountered in this science is, as I have already said elsewhere, that it has no means very suited to abbreviate the ideas and relations one has discovered. Thus, although it regulates the imagination and renders the mind just, it does not increase its extent by much and does not render it capable of discovering very compound truths.

But algebra, teaching one to abbreviate continually, and in the shortest manner in the world, ideas and their relations, increases extremely the capacity of the mind; for one cannot conceive anything so compound in the relations of magnitudes that the mind cannot, with time, discover it by the means it provides when one knows the way one must take in it.

The fifth rule and the others, where it is spoken of the manner of abbreviating ideas, pertain only to this science, for one has no convenient manner of abbreviating them in the other sciences. Thus I will not dwell on explaining them. Those who have much inclination for the mathematics and who wish to give their mind all the force and all the extent of which it is capable, and thus put themselves in a state to discover by themselves an infinity of new truths, having seriously applied themselves to algebra, will recognize that if this science is so useful for the search for truth, it is because it observes the rules we have prescribed. But I warn that by algebra I mean principally that of which M. Descartes and some others have made use.