Chapter Six

Being followed by Quinquisection (the opposite of Quintuplication) because the fifth part of the Subtended [Chord] is being sought.
We are able also for any Arc with a given Chord to find the Chord of the fifth part: but this has been found with a little more work, than had been with finding the third of the subtending chord. In both ways they are by Division, partially ordinary, partially Algebraic.

Let the Chord of 72 Degrees be given, or the line of the inscribed Pentagon 117557050458.

The places [positions] should be noted of the complete [fifth power] beyond the four from intermediate places, starting from the first place towards the left, namely from unity. Being noted also the places of the Cubes, inferred from the same principles, as you see here.

As this Chord really should subtend 72 Degrees and 288 Degrees, the Arc will give the Subtended fifth part both ways; surely of 14: 24' Degrees and 57: 36' Degrees. And if we should add on two whole Circles to the minor Arc, they will make 792 Degrees, of which the fifth part 158: 24' and of this Arc the Chord will be arrived at by the same equation: namely 5 - 5 + 1 = 117557050458. Because if we add a single circle to the Arcs 72: 0' and 288 :0' the sum of the degrees will be 432 and 648. And both the fifth parts of these will be arrived at, if the signs of the equations are changed thus: 5 - 5 - 1 = 117557050458. The Chords, I Say, of 86: 24' and 129: 36' shall be found: From the first, the Subtended Chord of 14: 24': being found. The first figure of the root being sought will be 2, of which the Cube 8, and of this the quintuple 40, because to its own place being added, as you see here, will increase the Dividend, making this 121557. From which ought to be taken away five times the root,

and the fifth power of the same root, being located in the correct places¹.

By which the first figure of the root sought will have been found, it will be fitting to think about correcting the Divisor². You see here the Divisor with its own places.

[By analogy:]
For the Cubic Gnomon being found by the Square of the root found tripled: of which five times will give the principal part five of the Cubic Gnomon being added to the Dividend [in the quintic].

We have in the right margin a way of finding Gnomons of fifth powers as of third powers, also have been added fourth powers, or biquadratics, and second powers, or Quadratics: because these lead together to the finding of fifth and third: which we apply only in the solution of equations.

¹ The first task is to find an approximate value of the smallest positive root, essentially by trial and error, of the 5^th power polynomial f(x) = -x⁵ + 5x³ - 5x + A, where A = 1.17557050458 = 2 sin 36. Briggs first evaluates f(x₁), where x₁ = 0.2, to find 0.2152505045, according to Table 6-1. Briggs always arranges things so that he subtracts positive quantities : he uses -f(x)/f'(x) for the smallest +ve root, and f(x)/ -f'(x) for the next one. See Figure 6-1 later in the text.

If x₁ is the first approximation to the required smallest +ve root, then a better approximation, according to Newton's method - though the method predates Newton a little as previously discussed in Note 4 of Chapter 3 - is furnished by x₂ = x₁ - f(x₁)/f'(x₁): and
where g₅(x₁, ) = 5x₁⁴ +10x₁³2 + 10x₁²³ + 5x₁³ + ⁵
g₄(x₁, ) = 4x₁³ +6x₁²2 + 10x₁³ + ⁴
g₃(x₁, ) = 3x₁² +3x₁2 + ³
while f'(x₁) = 5 - 15x₁² + 5 x₁⁴.

Thus, an 'ordinary' long division is performed, but only after the numerator or dividend has been adjusted by the correcting terms of the two gnomons called here g₅ and g₃ and the 5 terms, while the denominator or divisor is corrected by the g₄ term, and the last two terms in the divisor, which Briggs does not bother to write as the gnomon g₂. These operations constitute the algebraic part of the solution.

² This is an extended note on Table 6-3.

Table 6-2 is concerned with evaluating f'(x₁) as defined immediately above in Note 1, when x₁ = 0.2. The dividend is the first number in the top row, not used immediately in the calculation that follows, and the places are not aligned vertically with the numbers in the subsequent rows. These subsidiary calculations, of the nature of subroutines in modern computing, are designed to limit the arithmetic to the simplest operations possible. Note that some 22 pages will be needed to evaluate the root to the 22 places required to obtain a reference sine in the table of sines of adjacent angles used for interpolation to give the final tables of sines. It is appropriate in the present case to present Briggs' work in some detail, in order that the reader is left in no doubt as to what is going on. Later calculations will not be given in such detail, but the present example can be used to expand on them as required. We add the decimal point for our convenience.

Table [6-3A] appears to be a reference table of the different kinds of terms to be met with in evaluating the three kinds of gnomons, which share common factors, which need only be evaluated once.
We first enlarge on the gnomon of the 5^th power, g₅, that occupies the top right hand corner of Table 6-3:

Note that Briggs is only interested in the relative place of the contributions to his scheme at this stage, which he combines in an efficient way by considering all the terms as whole numbers [as he uses a radius of 10¹⁰ or larger]: thus, 5(x₁)⁴ gives 5 800000 = 4000000, etc. The connection between , set as 5, and 5(x₁), set as 100, gives internal consistency. There still remains the task of placing the leading number into the correct position in the decimal expansion. We next enlarge on the gnomon of the 4^th power, g₄, that occupies the middle right hand side of Table 6-3:

Note that the 6(x₁)²² term has been arrived at by adding two parts.
We next enlarge on the gnomon of the 3^rd power, g₃, that occupies the bottom right hand side of Table 6-3:

We are now in the position of tackling the main division algorithm, with the correcting factors being added to the numerator and denominator:

** The instruction 5 + 1 seems to have been placed inadvertently into this slot: the aim is to evaluate the gradient here, which is a slowly varying function .

We note that the ordinary division of the corrected dividend and divisor give the correction : 0.00 271894208/4.0820 = .000666. Hence, the new value of is 0.0006, while x₂ now has the value 0.250. Table 6-5 indicates how the new gnomon g₃(0.25,0.0006) = .000112770216.

In the same way, Table 6-5 gives the powers of 25 corresponding to the approximate root x₂ = 0.25, together with the powers of = 0.0006, and their various products, leading to g₅(0.25,0.0006) = 0. 000011767935. Note that the same divisor may be used without further change here, as only the first sig. fig. of the quotient is required: 0.0000271025225/4.0820 = 0.000066....

The dividend is further corrected, and the same divisor then gives: 0.0000026375347/4.0820 = 0.00000646... as the correction. The same divisor will fail eventually in supplying several correct consecutive figures and needs to be corrected again: so the process continues....

We can appreciate fully the nature of Briggs' work by resorting to a spreadsheet (which has the obvious advantage of setting out all the numbers in tables). For the present case, this has been done for 7 iterations, where the beautiful nature of the convergence becomes apparent, and one can appreciate Briggs' label of gnomon or pointer. For each correction has more correct figures, and after 7 iterations, the next 6 places are guaranteed correct: this was a powerful tool indeed for someone working by hand. However, rather than present the whole spreadsheet, only a few salient numbers are extracted from it:

Thus, the required root has the value 0.2506664671273953..

2. If from the same given Chord of 288 Degrees, with the fifth part of the Subtended Chord being sought: with the given points being noted above and below as before, and before, [i.e. to the right of], 5 - 5 + 1 , the Chord of 57: 36' being sought, 117557050458³.

The first figure will be 9, because by adding 5 the Dividend shall be 4820 [ 5(0.9)³ + 1.1755 = 4.82], whence the Divisor 5 can be taken away nine times. But if a lesser amount, five of the quotient cannot be taken away from the given number with the [lesser] increase of the five cubes. [f(.9) = 0.23, while f(0.8) = 0.59, and f(1) = -0.1755].

But the investigation itself is not different in any way from the preceding. For this given Chord and 5 [being added] (as they shall be the lesser in this operation with the main part as 5 and 1 ) ought to be taken from five of the side [root] and the fifth power of the side [root]: as thus,

³ We now look for a larger root of the 5^th power polynomial: for reference, the 5 real roots, and a graph of the function are included:

x₁ = -1.8096541049325596908121166953066;
x₂ = -1.3690942118564859775239691956940;
x₃ = .25066646712739535044210045155490;
x₄ = .96350734820450208227900164702114;
x₅ = 1.9645745014571482356149837924245.

Figure 6 - 1: Graph of the quintic f(x) = x⁵ - 5x³ + 5x - A
with the 5 real roots above, where A = 1.17557050458.

Note that Briggs avoids approaching a root from negative values; hence the first root found used the inverted function -f(x) as the numerator, but kept f'(x) as the denominator; while in the second case, f(x) is used, while -f'(x) is employed.

A spreadsheet analysis can be performed similar to that for the first root. Here we merely indicate the form of the correction factors as the iteration converges:

3. If we add two whole circles to 72 Degrees, the sum of the Degrees will be 792, the fifth part of which [will be] 158:24': Of which the Chord also should be given from the side [root] of the Pentagon: 5 - 5 + 1 = 117557050458

[Translator's Note: The rest of these tables, for the largest root above, and for the negative root considered next, are presented without explanation, as they follow the same scheme treated in great detail above for the first and second cases.]

4. If we add a single Circle to 288 Degrees; The sum will be 648, of which the fifth part 129: 36'. Thus the equation will be : 5 - 1 - 3 = 117557050458.