3. I shall now attempt the same by arguments drawn from the nature of the crookedness of the circle itself; but I shall first set down some premises necessary for this speculation; and
First, if a strait line be bowed into an arch of a circle equal to it, as when a stretched thread, which toucheth a right cylinder, is so bowed in every point, that it be everywhere coincident with the perimeter of the base of the cylinder, the flexion of that line will be equal, in all its points; and consequently the crookedness of the arch of a circle is everywhere uniform; which needs no other demonstration than this, that the perimeter of a circle is an uniform line.
Secondly, and consequently: if two unequal arches of the same circle be made by the bowing of two strait lines equal to them, the flexion of the longer line, whilst it is bowed into the greater arch, is greater than the flexion of the shorter line, whilst it is bowed into the lesser arch, according to the proportion of the arches themselves; and consequently, the crookedness of the greater arch is to the crookedness of the lesser arch, as the greater arch is to the lesser arch.
Thirdly: if two unequal circles and a strait line touch one another in the same point, the crookedness of any arch taken in the lesser circle, will be greater than the crookedness of an arch equal to it taken in the greater circle, in reciprocal proportion to that of the radii with which the circles are described; or, which is all one, any strait line being drawn from the point of contact till it cut both the circumferences, as the part of that strait line cut off by the circumference of the greater circle to that part which is cut off by the circumference of the lesser circle.
For let A B and A C (in the second figure) be two circles, touching one another, and the strait line A D in the point A; and let their centres be E and F; and let it be supposed, that as A E is to A F, so is the arch A B to the arch A H. I say the crookedness of the arch A C is to the crookedness of the arch A H, as A E is to A F. For let the strait line A D be supposed to be equal to the arch A B, and the strait line A G to the arch A C; and let A D, for example, be double to A G. Therefore, by reason of the likeness of the arches A B and A C, the strait line A B will be double to the strait line A C, and the radius A E double to the radius A F, and the arch A B double to the arch A H. And because the strait line A D is so bowed to be coincident with the arch A B equal to it, as the strait line A G is bowed to be coincident with the arch A C equal also to it, the flexion of the strait line A G into the crooked line A C will be equal to the flexion of the strait line A D into the crooked line A B. But the flexion of the strait line A D into the crooked line A B is double to the flexion of the strait line A G into the crooked line A H; and therefore the flexion of the strait line A G into the crooked line A C is double to the flexion of the same strait line A G into the crooked line A H. Wherefore, as the arch A B is to the arch A C or A H; or as the radius A E is to the radius A F; or as the chord A B is to the chord A C; so reciprocally is the flexion or uniform crookedness of the arch A C, to the flexion or uniform crookedness of the arch A H, namely, here double. And this may by the same method be demonstrated in circles whose perimeters are to one another triple, quadruple, or in whatsoever given proportion. The crookedness therefore of two equal arches taken in several circles are in proportion reciprocal to that of their radii, or like arches, or like chords; which was to be demonstrated.
Let the square A B C D be again described (in the third figure), and in it the quadrants A B D, B C A and D A C; and dividing each side of the square A B C D in the midst in E, F, G and H, let E G and F H be connected, which will cut one another in the centre of the square at I, and divide the arch of the quadrant A B D into three equal parts in K and L. Also the diagonals A C and B D being drawn will cut one another in I, and divide the arches B K D and C L A into two equal parts in M and N. Then with the radius B F let the arch F E be drawn, cutting the diagonal B D in O; and dividing the arch B M in the midst in P, let the strait line E a equal to the chord B P be set off from the point E in the arch E F, and let the arch a b be taken equal to the arch O a, and let B a and B b be drawn and produced to the arch A N in c and d; and lastly, let the strait line A d be drawn. I say the strait line A d is equal to the arch A N or B M.
I have proved in the preceding article, that the arch E O is twice as crooked as the arch B P, that is to say, that the arch E O is so much more crooked than the arch B P, as the arch B P is more crooked than the strait line E a. The crookedness therefore of the chord E a, of the arch B P, and of the arch E O, are as 0, 1, 2. Also the difference between the arches E O and E O, the difference between the arches E O and E a, and the difference between the arches E O and E b, are as 0, 1, 2. So also the difference between the arches A N and A N, the difference between the arches A N and A c, and the difference between the arches A N and A d, are as 0, 1, 2; and the strait line A c is double to the chord B P or E a, and the strait line A d double to the chord E b.
Again, let the strait line B F be divided in the midst in Q, and the arch B P in the midst in R; and describing the quadrant B Q S (whose arch Q S is a fourth part of the arch of the quadrant B M D, as the arch B R is a fourth part of the arch B M, which is the arch of the semiquadrant A B M) let the chord S e equal to the chord B R be set off from the point S in the arch S Q; and let B e be drawn and produced to the arch A N in f; which being done, the strait line A f will be quadruple to the chord B R or S e. And seeing the crookedness of the arch S e, or of the arch A c, is double to the crookedness of the arch B R, the excess of the crookedness of the arch A f above the crookedness of the arch A c will be subduple to the excess of the crookedness of the arch A c above the crookedness of the arch A N; and therefore the arch N c will be double to the arch c f. Wherefore the arch c d is divided in the midst in f, and the arch N f is ¾ of the arch N d. And in like manner if the arch B R be bisected in V, and the strait line B Q in X, and the quadrant B X Y be described, and the strait line Y g equal to the chord B V be set off from the point Y in the arch Y X, it may be demonstrated that the strait line B g being drawn and produced to the arch A N, will cut the arch f d into two equal parts, and that a strait line drawn from A to the point of that section, will be equal to eight chords of the arch B V, and so on perpetually; and consequently, that the strait line A d is equal to so many equal chords of equal parts of the arch B M, as may be made by infinite bisections. Wherefore the strait line A d is equal to the arch B M or A N, that is, to half the arch of the quadrant A B D or B C A.
Coroll. An arch being given not greater than the arch of a quadrant (for being made greater, it comes again towards the radius B A produced, from which it receded before) if a strait line double to the chord of half the given arch be adapted from the beginning of the arch, and by how much the arch that is subtended by it is greater than the given arch, by so much a greater arch be subtended by another strait line, this strait line shall be equal to the first given arch.
Supposing the strait line B V (in fig. 1) be equal to the arch of the quadrant B H D, and A V be connected cutting the arch B H D in I, it may be asked what proportion the arch B I has to the arch I D. Let therefore the arch A Y be divided in the midst in o, and in the strait line A D let A p be taken equal, and A q double to the drawn chord A o. Then upon the centre A, with the radius A q, let an arch of a circle be drawn cutting the arch A Y in r, and let the arch Y r be doubled at t; which being done, the drawn strait line A t (by what has been last demonstrated) will be equal to the arch A Y. Again, upon the centre A with the radius A t let the arch t u be drawn cutting A D in u; and the strait line A u will be equal to the arch A Y. From the point u let the strait line u s be drawn equal and parallel to the strait line A B, cutting M N in x, and bisected by M N in the same point x. Therefore the strait line A x being drawn and produced till it meet with B C produced in V, it will cut off B V double to B s, that is, equal to the arch B H D. Now let the point, where the strait line A V cuts the arch B H D, be I; and let the arch D I be divided in the midst in y; and in the strait line D C, let D z be taken equal, and D δ double to the drawn chord D y; and upon the centre D with the radius D δ let an arch of a circle be drawn cutting the arch B H D in the point n; and let the arch n m be taken equal to the arch I n; which being done, the strait line D m will (by the last foregoing corollary) be equal to the arch D I. If now the strait lines D m and C V be equal, the arch B I will be equal to the radius A B or B C; and consequently X C being drawn, will pass through the point I. Moreover, if the semicircle B H D ϐ being completed, the strait lines ϐ I and B I be drawn, making a right angle (in the semicircle) at I, and the arch B I be divided in the midst at i, it will follow that A i being connected will be parallel to the strait line ϐ I, and being produced to B C in k, will cut off the strait line B k equal to the strait line k I, and equal also to the strait line A γ cut off in A D by the strait line ϐ I. All which is manifest, supposing the arch B I and the radius B C to be equal.
But that the arch B I and the radius B C are precisely equal, cannot (how true soever it be) be demonstrated, unless that be first proved which is contained in art. 1, namely, that the strait lines drawn from X through the equal parts of O F (produced to a certain length) cut off so many parts also in the tangent B C severally equal to the several arches cut off; which they do most exactly as far as B C in the tangent, and B I in the arch B E; insomuch that no inequality between the arch B I and the radius B C can be discovered either by the hand or by ratiocination. It is therefore to be further enquired, whether the strait line A V cut the arch of the quadrant in I in the same proportion as the point C divides the strait line B V, which is equal to the arch of the quadrant. But however this be, it has been demonstrated that the strait line B V is equal to the arch B H D.
4. I shall now attempt the same dimension of a circle another way, assuming the two following lemmas.
Lemma I. If to the arch of a quadrant, and the radius, there be taken in continual proportion a third line Z; then the arch of the semiquadrant, half the chord of the quadrant, and Z, will also be in continual proportion.
For seeing the radius is a mean proportional between the chord of a quadrant and its semichord, and the same radius a mean proportional between the arch of the quadrant and Z, the square of the radius will be equal as well to the rectangle made of the chord and semichord of the quadrant, as to the rectangle made of the arch of the quadrant and Z; and these two rectangles will be equal to one another. Wherefore, as the arch of a quadrant is to its chord, so reciprocally is half the chord of the quadrant to Z. But as the arch of the quadrant is to its chord, so is half the arch of the quadrant to half the chord of the quadrant. Wherefore, as half the arch of the quadrant is to half the chord of the quadrant (or to the sine of 45 degrees), so is half the chord of the quadrant to Z; which was to be proved.
Lemma II. The radius, the arch of the semiquadrant, the sine of 45 degrees, and the semiradius, are proportional.
For seeing the sine of 45 degrees is a mean proportional between the radius and the semiradius; and the same sine of 45 degrees is also a mean proportional (by the precedent lemma) between the arch of 45 degrees and Z; the square of the sine of 45 degrees will be equal as well to the rectangle made of the radius and semiradius, as to the rectangle made of the arch of 45 degrees and Z. Wherefore, as the radius is to the arch of 45 degrees, so reciprocally is Z to the semiradius; which was to be demonstrated.
Let now A B C D (in fig. 4) be a square; and with the radii A B, B C and D A, let the three quadrants A B D, B C A and D A C, be described; and let the strait lines E F and G H, drawn parallel to the sides B C and A B, divide the square A B C D into four equal squares. They will therefore cut the arch of the quadrant A B D into three equal parts in I and K, and the arch of the quadrant B C A into three equal parts in K and L. Also let the diagonals A C and B D be drawn, cutting the arches B I D and A L C in M and N. Then upon the centre H with the radius H F equal to half the chord of the arch B M D, or to the sine of 45 degrees, let the arch F O be drawn cutting the arch C K in O; and let A O be drawn and produced till it meet with B C produced in P; also let it cut the arch B M D in Q, and the strait line D C in R. If now the strait line H Q be equal to the strait line D R, and being produced to D C in S, cut off D S equal to half the strait line B P; I say then the strait line B P will be equal to the arch B M D.
For seeing P B A and A D R are like triangles, it will be as P B to the radius B A or A D, so A D to D R; and therefore as well P B, A D and D R, as P B, A D (or A Q) and Q H are in continual proportion; and producing H O to D C in T, D T will be equal to the sine of 45 degrees, as shall by and by be demonstrated. Now D S, D T and D R are in continual proportion by the first lemma; and by the second lemma D C. D S:: D R. D F are proportionals. And thus it will be, whether B P be equal or not equal to the arch of the quadrant B M D. But if they be equal, it will then be, as that part of the arch B M D which is equal to the radius, is to the remainder of the same arch B M D; so A Q to H Q, or so B C to C P. And then will B P and the arch B M D be equal. But it is not demonstrated that the strait lines H Q and D R are equal; though if from the point B there be drawn (by the construction of fig. 1) a strait line equal to the arch B M D, then D R to H Q, and also the half of the strait line B P to D S, will always be so equal, that no inequality can be discovered between them. I will therefore leave this to be further searched into. For though it be almost out of doubt, that the strait line B P and the arch B M D are equal, yet that may not be received without demonstration; and means of demonstration the circular line admitteth none that is not grounded upon the nature of flexion, or of angles. But by that way I have already exhibited a strait line equal to the arch of a quadrant in the first and second aggression.
It remains that I prove D T to be equal to the sine of 45 degrees.
In B A produced let A V be taken equal to the sine of 45 degrees; and drawing and producing V H, it will cut the arch of the quadrant C N A in the midst in N, and the same arch again in O, and the strait line D C in T, so that D T will be equal to the sine of 45 degrees, or to the strait line A V; also the strait line V H will be equal to the strait line H I, or the sine of 60 degrees.
For the square of A V is equal to two squares of the semiradius; and consequently the square of V H is equal to three squares of the semiradius. But H I is a mean proportional between the semiradius and three semiradii; and, therefore, the square of H I is equal to three squares of the semiradius. Wherefore H I is equal to H V. But because A D is cut in the midst in H, therefore V H and H T are equal; and, therefore, also D T is equal to the sine of 45 degrees. In the radius B A let B X be taken equal to the sine of 45 degrees; for so V X will be equal to the radius; and it will be as V A to A H the semiradius, so V X the radius to X N the sine of 45 degrees. Wherefore V H produced passes through N. Lastly, upon the centre V with the radius V A let the arch of a circle be drawn cutting V H in Y; which being done, V Y will be equal to H O (for H O is, by construction, equal to the sine of 45 degrees) and Y H will be equal to O T; and, therefore, V T passes through O. All which was to be demonstrated.
I will here add certain problems, of which if any analyst can make the construction, he will thereby be able to judge clearly of what I have now said concerning the dimension of a circle. Now these problems are nothing else (at least to sense) but certain symptoms accompanying the construction of the first and third figure of this chapter.
Describing, therefore, again, the square A B C D (in fig. 5) and the three quadrants A B D, B C A and D A C, let the diagonals A C and B D be drawn, cutting the arches B H D and C I A in the middle in H and I; and the strait lines E F and G L, dividing the square A B C D into four equal squares, and trisecting the arches B H D and C I A, namely, B H D in K and M, and C I A in M and O. Then dividing the arch B K in the midst in P, let Q P the sine of the arch B P, be drawn and produced to R, so that Q R be double to Q P; and, connecting K R, let it be produced one way to B C in S, and the other way to B A produced in T. Also let B V be made triple to B S, and consequently, (by the second article of this chapter) equal to the arch B D. This construction is the same with that of the first figure, which I thought fit to renew discharged of all lines but such as are necessary for my present purpose.
In the first place, therefore, if A V be drawn, cutting the arch B H D in X, and the side D C in Z, I desire some analyst would, if he can, give a reason why the strait lines T E and T C should cut the arch B D, the one in Y, the other in X, so as to make the arch B Y equal to the arch Y X; or if they be not equal, that he would determine their difference.
Secondly, if in the side D A, the strait line D a be taken equal to D Z, and V a be drawn; why V a and V B should be equal; or if they be not equal, what is the difference.
Thirdly, drawing Z b parallel and equal to the side C B, cutting the arch B H D in c, and drawing the strait line A c, and producing it to B V in d; why A d should be equal and parallel to the strait line a V, and consequently equal also to the arch B D.
Fourthly, drawing e K the sine of the arch B K, and taking (in e A produced) e f equal to the diagonal A C, and connecting f C; why f C should pass through a (which point being given, the length of the arch B H D is also given) and c; and why f e and f c should be equal; or if not, why unequal.
Fifthly, drawing f Z, I desire he would show, why it is equal to B V, or to the arch B D; or if they be not equal, what is their difference.
Sixthly, granting f Z to be equal to the arch B D, I desire he would determine whether it fall all without the arch B C A, or cut the same, or touch it, and in what point.
Seventhly, the semicircle B D g being completed, why g I being drawn and produced, should pass through X, by which point X the length of the arch B D is determined. And the same g I being yet further produced to D C in h, why A d, which is equal to the arch B D, should pass through that point h.
Eighthly, upon the centre of the square A B C D, which let be k, the arch of the quadrant E i L being drawn, cutting e K produced in i, why the drawn strait line i X should be parallel to the side C D.
Ninthly, in the sides B A and B C taking g l and B m severally equal to half B V, or to the arch B H, and drawing m n parallel and equal to the side B A, cutting the arch B D in o, why the strait line which connects V l should pass through the point o.
Tenthly, I would know of him why the strait line which connects a H should be equal to B m; or if not, how much it differs from it.
The analyst that can solve these problems without knowing first the length of the arch B D, or using any other known method than that which proceeds by perpetual bisection of an angle, or is drawn from the consideration of the nature of flexion, shall do more than ordinary geometry is able to perform. But if the dimension of a circle cannot be found by any other method, then I have either found it, or it is not at all to be found.
From the known length of the arch of a quadrant, and from the proportional division of the arch and of the tangent B C, may be deduced the section of an angle into any given proportion; as also the squaring of the circle, the squaring of a given sector, and many the like propositions, which it is not necessary here to demonstrate. I will, therefore, only exhibit a strait line equal to the spiral of ArchimedesArchimedes, and so dismiss this speculation.
5. The length of the perimeter of a circle being found, that strait line is also found, which touches a spiral at the end of its first conversion. For upon the centre A (in fig. 6) let the circle B C D E be described; and in it let Archimedes' spiral A F G H B be drawn, beginning at A and ending at B. Through the centre A let the strait line C E be drawn, cutting the diameter B D at right angles; and let it be produced to I, so that A I be equal to the perimeter B C D E B. Therefore I B being drawn will touch the spiral A F G H B in B; which is demonstrated by Archimedes in his book De Spiralibus.
And for a strait line equal to the given spiral A F G H B, it may be found thus.
Let the strait line A I, which is equal to the perimeter B C D E, be bisected in K; and taking K L equal to the radius A B, let the rectangle I L be completed. Let M L be understood to be the axis, and K L the base of a parabola, and let M K be the crooked line thereof. Now if the point M be conceived to be so moved by the concourse of two movents, the one from I M to K L with velocity encreasing continually in the same proportion with the times, the other from M L to I K uniformly, that both those motions begin together in M and end in K; Galilæus has demonstrated that by such motion of the point M, the crooked line of a parabola will be described. Again, if the point A be conceived to be moved uniformly in the strait line A B, and in the same time to be carried round upon the centre A by the circular motion of all the points between A and B; Archimedes has demonstrated that by such motion will be described a spiral line. And seeing the circles of all these motions are concentric in A; and the interior circle is always less than the exterior in the proportion of the times in which A B is passed over with uniform motion; the velocity also of the circular motion of the point A will continually increase proportionally to the times. And thus far the generations of the parabolical line M K, and of the spiral line A F G H B, are like. But the uniform motion in A B concurring with circular motion in the perimeters of all the concentric circles, describes that circle, whose centre is A, and perimeter B C D E; and, therefore, that circle is (by the coroll. of art. 1, chap, XVI) the aggregate of all the velocities together taken of the point A whilst it describes the spiral A F G H B. Also the rectangle I K L M is the aggregate of all the velocities together taken of the point M, whilst it describes the crooked line M K. And, therefore the whole velocity by which the parabolical line M K is described, is to the whole velocity with which the spiral line A F G H B is described in the same time, as the rectangle I K L M is to the circle B C D E, that is to the triangle A I B. But because A I is bisected in K, and the strait lines I M and A B are equal, therefore the rectangle I K L M and the triangle A I B are also equal. Wherefore the spiral line A F G H B, and the parabolical line M K, being described with equal velocity and in equal times, are equal to one another. Now, in the first article of chap. XVIII, a strait line is found out equal to any parabolical line. Wherefore also a strait line is found out equal to a given spiral line of the first revolution described by Archimedes; which was to be done.
6. In the sixth chapter, which is of Method, that which I should there have spoken of the analytics of geometricians I thought fit to defer, because I could not there have been understood, as not having then so much as named lines, superficies, solids, equal and unequal, &c. Wherefore I will in this place set down my thoughts concerning it.
Analysis is continual reasoning from the definitions of the terms of a proposition we suppose true, and again from the definitions of the terms of those definitions, and so on, till we come to some things known, the composition whereof is the demonstration of the truth or falsity of the first supposition; and this composition or demonstration is that we call Synthesis. Analytica, therefore, is that art, by which our reason proceeds from something supposed, to principles, that is, to prime propositions, or to such as are known by these, till we have so many known propositions as are sufficient for the demonstration of the truth or falsity of the thing supposed. Synthetica is the art itself of demonstration. Synthesis, therefore, and analysis, differ in nothing, but in proceeding forwards or backwards; and Logistica comprehends both. So that in the analysis or synthesis of any question, that is to say, of any problem, the terms of all the propositions ought to be convertible; or if they be enunciated hypothetically, the truth of the consequent ought not only to follow out of the truth of its antecedent, but contrarily also the truth of the antecedent must necessarily be inferred from the truth of the consequent. For otherwise, when by resolution we are arrived at principles, we cannot by composition return directly back to the thing sought for. For those terms which are the first in analysis, will be the last in synthesis; as for example, when in resolving, we say, these two rectangles are equal, and therefore their sides are reciprocally proportional, we must necessarily in compounding say, the sides of these rectangles are reciprocally proportional, and therefore the rectangles themselves are equal; which we could not say, unless rectangles have their sides reciprocally proportional, and rectangles are equal, were terms convertible.
Now in every analysis, that which is sought is the proportion of two quantities; by which proportion, a figure being described, the quantity sought for may be exposed to sense. And this exposition is the end and solution of the question, or the construction of the problem.
And seeing analysis is reasoning from something supposed, till we come to principles, that is, to definitions, or to theorems formerly known; and seeing the same reasoning tends in the last place to some equation, we can therefore make no end of resolving, till we come at last to the causes themselves of equality and inequality, or to theorems formerly demonstrated from those causes; and so have a sufficient number of those theorems for the demonstration of the thing sought for.
And seeing also, that the end of the analytics is either the construction of such a problem as is possible, or the detection of the impossibility thereof; whensoever the problem may be solved, the analyst must not stay, till he come to those things which contain the efficient cause of that whereof he is to make construction. But he must of necessity stay, when he comes to prime propositions; and these are definitions. These definitions therefore must contain the efficient cause of his construction; I say of his construction, not of the conclusion which he demonstrates; for the cause of the conclusion is contained in the premised propositions; that is to say, the truth of the proposition he proves is drawn from the propositions which prove the same. But the cause of his construction is in the things themselves, and consists in motion, or in the concourse of motions. Wherefore those propositions, in which analysis ends, are definitions, but such as signify in what manner the construction or generation of the thing proceeds. For otherwise, when he goes back by synthesis to the proof of his problem, he will come to no demonstration at all; there being no true demonstration but such as is scientifical; and no demonstration is scientifical, but that which proceeds from the knowledge of the causes from which the construction of the problem is drawn. To collect therefore what has been said into few words; ANALYSIS is ratiocination from the supposed construction or generation of a thing to the efficient cause or coefficient causes of that which is constructed or generated. And SYNTHESIS is ratiocination from the first causes of the construction, continued through all the middle causes till we come to the thing itself which is constructed or generated.
But because there are many means by which the same thing may be generated, or the same problem be constructed, therefore neither do all geometricians, nor doth the same geometrician always, use one and the same method. For, if to a certain quantity given, it be required to construct another quantity equal, there may be some that will inquire whether this may not be done by means of some motion. For there are quantities, whose equality and inequality may be argued from motion and time, as well as from congruence; and there is motion, by which two quantities, whether lines or superficies, though one of them be crooked, the other strait, may be made congruous or coincident. And this method Archimedes made use of in his book De Spiralibus. Also the equality or inequality of two quantities may be found out and demonstrated from the consideration of weight, as the same Archimedes did in his quadrature of the parabola. Besides, equality and inequality are found out often by the division of the two quantities into parts which are considered as indivisable; as Cavallerius Bonaventura has done in our time, and Archimedes often. Lastly, the same is performed by the consideration of the powers of lines, or the roots of those powers, and by the multiplication, division, addition, and subtraction, as also by the extraction of the roots of those powers, or by finding where strait lines of the same proportion terminate. For example, when any number of strait lines, how many soever, are drawn from a strait line and pass all through the same point, look what proportion they have, and if their parts continued from the point retain everywhere the same proportion, they shall all terminate in a strait line. And the same happens if the point be taken between two circles. So that the places of all their points of termination make either strait lines, or circumferences of circles, and are called plane places. So also when strait parallel lines are applied to one strait line, if the parts of the strait line to which they are applied be to one another in proportion duplicate to that of the contiguous applied lines, they will all terminate in a conical section; which section, being the place of their termination, is called a solid place, because it serves for the finding out of the quantity of any equation which consists of three dimensions. There are therefore three ways of finding out the cause of equality or inequality between two given quantities; namely, first, by the computation of motions; for by equal motion, and equal time, equal spaces are described; and ponderation is motion. Secondly, by indivisibles: because all the parts together taken are equal to the whole. And thirdly, by the powers: for when they are equal, their roots also are equal; and contrarily, the powers are equal, when their roots are equal. But if the question be much complicated, there cannot by any of these ways be constituted a certain rule, from the supposition of which of the unknown quantities the analysis may best begin; nor out of the variety of equations, that at first appear, which we were best to choose; but the success will depend upon dexterity, upon formerly acquired science, and many times upon fortune.
For no man can ever be a good analyst without being first a good geometrician; nor do the rules of analysis make a geometrician, as synthesis doth; which begins at the very elements, and proceeds by a logical use of the same. For the true teaching of geometry is by synthesis, according to Euclid's method; and he that hath Euclid for his master, may be a geometrician without Vieta, though Vieta was a most admirable geometrician; but he that has Vieta for his master, not so, without Euclid.
And as for that part of analysis which works by the powers, though it be esteemed by some geometricians, not the chiefest, to be the best way of solving all problems, yet it is a thing of no great extent; it being all contained in the doctrine of rectangles, and rectangled solids. So that although they come to an equation which determines the quantity sought, yet they cannot sometimes by art exhibit that quantity in a plane, but in some conic section; that is, as geometricians say, not geometrically, but mechanically. Now such problems as these, they call solid; and when they cannot exhibit the quantity sought for with the help of a conic section, they call it a lineary problem. And therefore in the quantities of angles, and of the arches of circles, there is no use at all of the analytics which proceed by the powers; so that the ancients pronounced it impossible to exhibit in a plane the division of angles, except bisection, and the bisection of the bisected parts, otherwise than mechanically. For Pappus, (before the 31st proposition of his fourth book) distinguishing and defining the several kinds of problems, says that "some are plane, others solid, and others lineary. Those, therefore, which may be solved by strait lines and the circumferences of circles, (that is, which may be described with the rule and compass, without any other instrument), are fitly called plane; for the lines, by which such problems are found out, have their generation in a plane. But those which are solved by the using of some one or more conic sections in their construction, are called solid, because their construction cannot be made without using the superficies of solid figures, namely, of cones. There remains the third kind, which is called lineary, because other lines besides those already mentioned are made use of in their construction, &c." And a little after he says, "of this kind are the spiral lines, the quadratrices, the conchoeides, and the cissoeides, And geometricians think it no small fault, when for the finding out of a plane problem any man makes use of conics, or new lines." Now he ranks the trisection of an angle among solid problems, and the quinquesection among lineary. But what! are the ancient geometricians to be blamed, who made use of the quadratrix for the finding out of a strait line equal to the arch of a circle? And Pappus himself, was he faulty, when he found out the trisection of an angle by the help of an hyperbole? Or am I in the wrong, who think I have found out the construction of both these problems by the rule and compass only? Neither they, nor I. For the ancients made use of this analysis which proceeds by the powers; and with them it was a fault to do that by a more remote power, which might be done by a nearer; as being an argument that they did not sufficiently understand the nature of the thing.
The virtue of this kind of analysis consists in the changing and turning and tossing of rectangles and analogisms; and the skill of analysts is mere logic, by which they are able methodically to find out whatsoever lies hid either in the subject or predicate of the conclusion sought for. But this doth not properly belong to algebra, or the analytics specious, symbolical, or cossick; which are, as I may say, the brachygraphy of the analytics, and an art neither of teaching nor learning geometry, but of registering with brevity and celerity the inventions of geometricians. For though it be easy to discourse by symbols of very remote propositions; yet whether such discourse deserve to be thought very profitable, when it is made without any ideas of the things themselves, I know not.
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Fig. 1-2
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Fig. 5-6