Chapter Nine

With the powers of these Chords, and altogether everything prepared in this way, we will be able to test with any of the sections demonstrated or talked about. And for any Chord being given, we will be able to find the Chord for any multiples of the Arc, or fractions of the Arc given.

With multiples all being easily expounded; with fractions the thing is not less certain, however it is with a lot more working. We saw the method for Trisection and Quinquisection for these parts of Chords; Now for Squares with Bisection, and [then] we shall see to the rest.

With the equations being serviced by Squares [i.e. even powers], they will be able to be reduced to more suitable terms, and from the Units¹ [constant term] less the more removed [even powers]; which meet [i.e. are adjacent] on account of the equality of the intervals, [i.e. there are only even powers, going up in steps of 2] which is always the case for the powers serving these equations. As with Bisection, the number of units being equal to 4 - 1 , where there are two intervals, between unity and , and the same total between and : thus with all the rest from which Squares are being found. But with the equations for the Chords themselves the intervals are always unequal [i.e. those we have already met]. As with Trisection, the units [constant term] being equal to 3 - 1 .

Between the 0 power and there is a single interval; But between and there are two intervals. Because of this for Bisection for we will be able to put ; for the way after completing the operation we understand to have found the root by reverting that Square which was being sought: of which the root will be the Chord. E.g. the Square [of the Chord] subtending 90 degrees is 2. The Square subtending 45 degrees being sought. I assert 2 = 4 - 1 , and by making a reduction 2 = 4 - 1 .

The value of the root being sought we will be able to find by two methods: With one being general, which agrees for all equations, (which before we made use of in finding these Chords themselves, for Trisection and Quinquisection): namely by dividing the given number of the power by the number of the side, and between Dividing the Quotient, the Square found should be added on [the gnomon]. With the other special [method], which agrees with Bisection to such an extent, where as many as three powers are being compared between themselves: which is the common method expounded by writers of Arithmetic [i.e. the usual method of solving a quadratic by completing the square].

Particularly we should make use of the general method. Let the given Square of the Chord of 140 Degrees be 3532088886237956070404 The Square of the Chord of 70:0': Degrees being sought. The equation of the Chord being agreed upon 4 - 1 , which being reduced to 4 - 1.



Therefore by this method we have found the value of the side² [ we shall call this the root henceforth] sought 13159597133486625339118. But this root is not the Chord of 70:0' Degrees but the Square of the same Chord, as warned before.

3. The same value of the root being found by the usual method in most of the works of the writers of Arithmetic. If the roots being equal to the units and the Square¹: If from the Square of half the number of the root being taken away the units: the root of the remainder either taken or added to the same half shall give the value of the root sought.
[Thus, if bx = x² + A, x = b/2 ((b/2)² - A)].

Given the equation 4 being equal to 1 + 3532088886237956070404 . The Square of half the number of the roots 4: from which if being taken the units there will remain
0467911113762043929595214, of which the root 06840402866513374660882 which being subtracted from half the number of the roots 2, there will remain 13159597133486625339118, the Square of the Chord of 70:0'. For if to the root found 06840402866513374660882 being added to half of the number of the roots, the sum will be 26840402866513374660882 the Square [of the Chord] subtending 110:0' to the complementary Arc of the Semicircle.

And both of these equations satisfy the given equations.

¹ Briggs designates the terms of a polynomial a + bx + cx² + dx³ + ex⁴ + ... according to the following scheme: the constant term a he calls 'the units', from unitas; the linear term x is the root or side, from latus, while b is the number of the root, etc; x² is the square or quadratic term; x³ the cubic term, x⁴ the biquadratic, x⁵ the quintic term, and so on.

² The working of Table 9-1 is presented here for the first 10 approximations.

As the denominator is approximately constant, and as the leading term only in the division was required, a lot of the working was mental, and so does not appear in Briggs' table.

Let the Square of the Chord of 36:0': be given. Being sought the Square of the Chord of 18:0': from the equation 4 - 1 = 038196601125010515176.

Therefore the Square being sought is 0097886967409692855.
It will come out the same by the other method¹.

4 = 1 + 038196601125010515176. Half the number of the root is 2.
4 . . . . . . . . . . . . . The Square of half the number of the root.
361803398874989484824 The Difference: of which the root is
190211303259030714 which if being taken and added to the two, the half the number of the root, will come to 0097886967409692855 the Square of the Chord of 18 Degrees: and 390211303259030714 the Square of the Chord of 162 Degrees, the compliment of course to the semicircle: the other [root] which will satisfy the equation given.

Therefore with Bisection we will be able to make use of either the general method, which will agree for all equations, or of the common method, which will agree only for these equations for which three different kinds of figures equidistant being equal among themselves. [ i.e. the relation can be cast as a quadratic; the other method obviously works for all degrees of polynomials.]

4. Because if either there will have been more kinds, or the intervals shall be unequal, we ought to have recourse to that more general method. As with Trisection, if the Square of the Third of the Chord is sought: 9 - 6 + 1 being equal to the Square of the Chord: and by the factor reduction 9 - 6 + 1 being equal to the given Square.

Let the given Square of the Chord be, 72:0': 1381966011250105152. Being sought the Square of the Chord 24:0': the Equation is3 : 9 - 6 + 1 = 1381966.

³

Table 9-5 contains a fragment of the calculations of the root, the reader should be able to find what the various numbers mean either from the brief notes inserted into the table itself, or from this adjoining spreadsheet calculation, Table 9-5A. The work follows from the extensive discussion in Chapter Four.