Bisect the primary arc FM in the point m, and produce the chord Fm of the half arc Fm, through m, to intercept the transverse arc MK in the point e. Join Am. Then because the chord FM is perpendicular to the radius A m, and the chord Fm perpendicular to the radius which bisects the angle FA m, the line Fm e bisects the transverse angle MF K. Bisect the half are Fm in the point j, and through j, from F, draw Fjf intercepting the transverse arc MK in f. Join Aj. Then because radius Aj bisects the angle F Am and is perpendicular to the line Fme, and the line Fjf is perpendicular to the radius which bisects the angle F Aj, the line Fƒ bisects the transverse angle e F K.

Wherefore it is made evident that whatever part of the primary arc be cut off by a chordline drawn from its original point, if that chord-line be produced to intercept the transverse arc, the part of the transverse arc cut off thereby and contained between the point of interception and the tangent line of the primary, must have the same ratio to the whole transverse arc, which the part, cut off by the chord-line from the primary, has to the whole primary arc.

PART THIRD.

"QUADRATRIX" AND "LUNAR ANALYSIS OF THE CIRCLE;"

WITH LUNAR PROBLEMS.

THE QUADRATRIX OF DINOSTRATUS:

We will first quote the account of the Quadratrix by Professor Wallace in the Encyclopædia Britannica.

("Squaring or Quadrature of the Circle ")

"We learn from Simplicius that Nicomedes and Apollonius both attempted to square the circle, the former by means of a curve which he called the quadratrix; the invention of which, however, is ascribed to Dinostratus ; and the latter, by the help of a curve, denominated the sister to the tortuous line, or spiral, and which was probably the quadratrix of Dinostratus ; the nature of which and the manner of its application to the subject in question, we shall briefly explain.

A

FIG. 2.

"Let AFB be a quadrant of a circle (Fig. 2), and Cits centre; and conceive the radius CF to revolve uniformly about C, from the position CA, until at last it coincides with CB; while at the same time a line D G is carried with a uniform motion from A towards CB; the former line continuing always parallel to the latter until at last they coincide; both motions being supposed to begin and end at the same instant: The point in which the revolving radius C F and the moveable line DG intersect one another, will generate a certain curve line A E H, which is the quadratrix of Dinostratus.

D

E

KH L

R

"Draw EK, FL, both perpendicular to CB; then because the radius AC, and the quadrantal arch AFB, are uniformly generated in the same time by the points D and F, the contemporaneous spaces described will have to one another the same ratio as the whole spaces; that is AD: A F:: A C:A B; hence we have A C: A B :: DC, or E K, : F B. Now, as the moveable point F approaches to B, the ratio of the straight line E K to the arch FB will approach to, and will manifestly be ultimately the same as, the ratio of the straight line E K to the straight line FL, which again is equal to the ratio of CE to CF; therefore the ratio of the radius AC to the quadrantal arch AFB is the limit of the ratio of CE to CF, and consequently equal to the ratio of CH to CB; I being the point in which the quadratrix meets C B. Since, therefore CH: CB:: CA, or C B, : quad. arch A FB, if by any means we could determine the point H, we might then find a straight line equal to the quadrantal arch (by finding a third proportional to C H and C B), and consequently a straight line equal

Then, because (by

case) ad is pro

D

H

(Vig Beat the perpendicular radis AC in d. From d, at right angles to A C, draw dg, intersecting the quadratriz in g. Bisect the quadrantal are A B in J. Join g J, and join also dJ, cutting the qua/iratrix in L. From L, at right angles to AC, draw fa, intercepting A Cin a, And through f, from C, draw the radius Cb, intercepting the quad. arc in b; also, join DF, cutting the quadratrix in g. the synchronal conditions of the portional to bJ in the same ratio as Ad: AJ, and as AC: AB, we have af: fb : ; dg : g J::df: ƒ J. And since d D is also proportional to J F in the ratio of AO AB, we have also Dg:gF::df:fJ. And, finally, dg: gJ::df:fJ::Dg: "/ F: CH: HB. Evidently, therefore, if the perpendicular A C′ and the arc A B be proportionately divided into any number of parts, so that each successive part of A B be proportional to the corresponding successive part of AC in the ratio of the whole of A B to the whole of 4 C, and then the successive divisional points in A C be directly joined to the corresponding successive points in the arc A B, that part of each connecting line, between the quadratrix curve and the perpendicular 4 C, will be proportional to the

* No far from the assumption "Now as the moveable point Fapproaches to B, the ratio of the straight line ZK to the are F' I will approach to and ultimately become the same as the ratio of the straight line E K to the straight line FL" being manifestly true, it seems to us to be manifestly untrue with regard to the latter part of it, for the sine of an arc and the arc itself can never become coincident, nor can the ratio of any fraction of the lesser sine EK to a similar fraction of the greater sine F1, ever become equal to the ratio of the former to a similar fraction of the greater are AFB,

↑ Moo Primary Definitions, No. 5, pago 14.

other part of the connecting line, between the quadratrix and the arc A B, in the ratio of CH to HB; (as, for example df:fJ and Dg:g F). Hence it follows (1) that the ratio of CH to HB is not, as Professor Wallace supposes, a derivative from and ultimate limit to the ratio of CE to E F, for it exists from the very commencement and remains constant throughout the description of the quadratrix. And (2) because the ratio (of CH to H B) remains constant from the commencement throughout the quadratrix curve, that curve is so uniformly related to the quadrantal arc A B that, as AB is a cyclal curve, so must the quadratrix be cyclal also; for, manifestly, if the curvature of AH were in one part greater than or equal to, and in another part less than, cyclal, it would be impossible that the parts of the connecting lines cut off by the quadratrix, on the one side of it, could have the same uniform constant ratio to the corresponding remaining parts of those lines, on the other side of it, throughout the curve. Wherefore it clearly appears that the quadratrix A H is a simple cyclal curve; i. e., the arc of a circle.

Having determined the quadratrix curve AH to be cyclal, by application of Prop. 39 (P.F.), its centre of description may be at once determined. Accordingly (Fig. 4):-Draw the chord AH, and bisect it in

n

FIG. 4.

the point n. Produce BC through C, and from n, at right angles to AH, draw n R, intercepting the production of B C in the point R. Then R is the centre of description and RA the radius of the arc (quadratrix) A H. The trigonometrical measurement of the radius, and determination of the value of CH, readily presents itself as follows:-Draw the chord Ag, of the half quadratrix, and bisect it in p; and, at right angles to A C, draw pq. Join p R cutting A Cin its point of bisection d. Then since qp and q d are manifestly equal, d C (the half of A C) equals CR. Therefore A C equals twice RC and RA is the diagonal of the rectangle contained by AC and CR. Now radius RH is the same as RA; and CH=RH-RC. Let A C=1. Then RC=5. RA (or RH) = 1·118034... CH = 1·118034... CH618034... Let CH=1. Then CB= 1.618034...* Hence, if the ratio CB: CH equalled the ratio of the quadrantal arc

RA

=

R

C

H

B

We have here a very interesting numerical (quantitative) relationship. The greater radius ✓1·25 = 1·118034... From this deduct 5. Then the prime number 1.0 is a mean proportional between the remainder and that remainder plus 10 (namely 618034: 1·0 :: 1·0:1·618034), as appears above. In the same connexion it may be remarked as noteworthy that the angle subtended by the quadratrix is nearly a mean proportional between 45° and 90°. For the mean would be (about) 63°-64 and the actual angle is (about) 63°44. This latter circumstance, however, is (pro

P

to its radius, (1.61803 x 2)=3.23606... would be the ratio of the circle's circumference to the diameter as unity, but which is certainly much greater (by nearly one part in thirty-three) than the true ratio.

Although it now appears that the quadratrix does not possess that extraordinary property first attributed to it by its discoverer, Dinostratus, in ancient tires, and also assigned to it by Prof. Wallace and other mathematicians up to the present time, we need not conclude that its actual relationship to the quadrantal are is devoid of interest to geometricians. Before passing on, we will observe (1) that if a point be taken anywhere between C and R, for a centre of description, and, joining that point with A for a radius, an are be described therewith terminated by the line R CB, that are will be intermediate between the quadratrix and the quadrantal arc. Consequently, by taking any number of successive points, between C and R, a corresponding number of arcs may be described between the quadratrix and quadrantal arc-all of them having the point A in common (at the original extremity of each). In like manner, by taking a point as the centre of description further than R from C, an arc, having the same point A at its original extremity, may be described between the quadratrix and the perpendicular A C; and so, by taking successive points further than R from C, any number of arcs, having the point A in common, may be described between the quadratrix and the perpendicular 4 C. The limit to this process shows the perpendicular AC to have a twofold character, for, whilst it is the radius of the quadrantal are, it may be also considered as the ultimate quadratrix when the centre of description, beyond R, becomes infinitely distant and the radius of infinite length: for then (if it were possible), no curvature remaining, the quadratrix would become coincident with the straight line. (2.) That the uniformity of the ratios, in the parts of the lines directly connecting the divisional points in the perpendicular radius AC with the corresponding divisional points in the quadrantal arc A B, on each side respectively of the quadratrix, holds good with regard to each of the other intermediate arcs, whether nearer to or further from the quadrantal arc than the quadratrix itself, and for the same reason, namely—that in each of them the curvature is cyclal.

The angle subtended by the quadratrix=about 63°-435. Since it may be considered as areally belonging either to the lesser circle-of which CA is the radius, or to its own circle-of which RA is the radius, the area of the figure contained by the right angle ACB and the quadratrix may be expressed in terms of either radius. Thus as a part of the lesser circle (when CA=1) the area of the figure=4422...; as a part of its own circle (when RA=1) its area=35369... [This is reckoning the angle, as stated, 63°-435; and the circle's area as 3·14269...]

bably-so to speak) merely a coincidence, but the former is a fact belonging to the natural correlations of numbers, quantities and magnitudes.