A treatise of angular sections by John Wallis ...
Wallis, John, 1616-1703., Wallis, John, 1616-1703. Treatise of algebra.
Page  60

### CHAP. VIII. Of the Canon of Subtenses, and Sines; Of Tangents also and of Secants.

FROM what is delivered in the foregoing Chapter; it is easie to con∣struct a Canon of Subtenses or Chords, in Surd Roots, to every Three∣halves of a Degree throughout the Semicircle. The halves of which Sub∣tenses, are the Right-sines for every Three-quarters of a Degree throughout the Quadrant.

(And thence, if need be, many Canons of Tangents and Secants, be deduced, in Surds Roots.)

For to every Subtense, to be successively sought, there will need but one extraction of the Square Root; (and, sometimes, not this;) the rest of the work being dispatched by only Addition and Subtraction; or, at most, Division also by 2 or 4.

As, for Example: Supposing the Radius of a Circle R = 1. Then (because these, in the same Circle be all equal,) C = D = 1. And likewise Cq = Dq = CD = 1. And B will be the Subtense of the Angle proposed.

Therefore, (by § 1.) the Square of the Subtense of 90 Degrees, Bq = Cq + Dq = 1 + 1 = 2. And the Subtense it self 〈 math 〉: Which is had by one extraction of the Square Root of the number 2; continued in Decimal Parts to to what accuracy we please. Suppose 〈 math 〉proximè.

Again, (by § 2.) the Square of the Subtense of 120 Degrees, is Bq = Cq + Dq + CD = 1 + 1 + 1 = 3. And the Subtense it self 〈 math 〉: Which is likewise had by one extraction of the Square Root of 3. Suppose 〈 math 〉proximè.

The Square of the Subtense of 60 Degrees, is (by § 3.) Bq = Cq + Dq − CD = 1 + 1 − 1 = 1. And therefore the Subtense B = 1.

The Square of the Subtense of 135 Degrees, is (by § 4.) 〈 math 〉. Which is had by adding 2, to the value of 〈 math 〉 already found at § 1. That is, 〈 math 〉. And so by one extraction of the Square Root of this number, we have the Subtense of 135 Degrees: Namely, 〈 math 〉proximè.

So (by § 5.) the Square of the Subtense of 45 Degrees, is 〈 math 〉. That is (by subtracting from 2 the value of 〈 math 〉 already found,) Bq = 0.58578643 ½ proximè. The Square Root of which, now to be extracted, is 〈 math 〉proximè.

Page  61And (by § 6.) the Square of the Subtense of 150 Degrees, is 〈 math 〉. Which is had, by adding 2 to the value of 〈 math 〉 all ready found. That is, 〈 math 〉. The Root of which (now to be extracted) is 〈 math 〉proximè.

Or thus: Because 〈 math 〉, (as will appear, either by the Squaring of this, or by extracting the Square Root of the Binomial 〈 math 〉) Having, as before, the value of 〈 math 〉; and (by one extraction now to be made) the value of 〈 math 〉 (or, it may be had by Multiplying the value of 〈 math 〉, by that of 〈 math 〉, already known; because 〈 math 〉) we have thence 〈 math 〉; and the half thereof 〈 math 〉. As before,

So (by § 7.) the Square of the Subtense of 30 Degrees, is 〈 math 〉, (which is had by Subduction only, the value of 〈 math 〉 being found before.) The Square Root of which (now to be extracted) is the Subtense 〈 math 〉proximè.

Or thus, (without extracting a Root;) because 〈 math 〉: Therefore, (the values of 〈 math 〉 and 〈 math 〉 being had before:) by Subduction only we have 〈 math 〉proximè; and (the half of this) 〈 math 〉proximè: As before,

And in the rest, (taking the Propositions or Paragraphs as they are before set down in the former Chapter,) there is need but of one extraction of the Square Root (and oft-times not of one,) for finding of each Subtense.

These Subtenses being thus had; the halves thereof are the Right-sines of the half Arch. As for Example.

 Arches. Subtenses. Sines. Arches. Degrees. 90 1.41421356 + 0.70710678 + 45 Degrees. Degrees. 120 1.73205081 − 0.86602540 ½ 60 Degrees. Degrees. 60 1.00000000 0.50000000 30 Degrees. Degrees. 135 1.84775906 ½ 0.92387953 ¼ 67 ½ Degrees. Degrees. 45 0.76536687 − 0.38268343 ½ 22 ½ Degrees. Degrees. 150 1.93185165 + 0.96592582 ½ 75 Degrees. Degrees. 30 0.51763809 0.25881904 ½ 15 Degrees.

Now follows the Table of Subtenses in Surd Roots, answering to each three halves of a Degree throughout the whole Semicircle; (and consequently, of their Residuals to a whole Circle, whose Subtenses are the same with these:) Putting the Radius R = 1, and therefore C = D = 1, and likewise Cq = Dq = CD = 1: With references in the Margin to the Paragraphs of the for∣mer Chapter from whence they are derived.

Page  62〈 math 〉Page  63〈 math 〉Page  64〈 math 〉Page  65〈 math 〉Page  66 And, in like manner, we may proceed to design, by Surd Roots, the Sub∣tenses of Arches as small as we please, by a continual Bisection of these Arches. The halves of which Subtenses, are the Right Sines of the Half-Arches.

But, to design an intire Canon of Subtenses and Sines, answering to each single Degree, and the Sexagesims or first Minutes of such Degrees: Will (beside the extracting the Square Roots, of such Surds, in Numbers,) require also the Analysis (in Numbers) of Two Trisections, and of one Quinquisection of an Arch.

For, the former process reaching no farther than to the Subtense of 1½ De∣gree; and consequently to the sine of ¾ of a Degree, or of Min. 45 = 3 × 3 × 5: We may thence, by a Trisection twice performed; and a Quinquisection once, proceed to the sine of 1 Minute. But not by Bisections only, or operations thence deduced.

But, these operations being so (as is said) performed; the rest of the work is easily dispatched by help of § 9, 10. Chap. 6. for finding the Subtense of the Sum or Difference of those Arches whose Subtenses are already known.