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 Archimedes[Archimedes], and so dismiss this speculation.
The equation of the spiral of Archimedes with a strait line.
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.
Of the analysis of geometricians by the powers of lines.
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.