1. With the Sines of any Arcs of equal difference, being found by this method, the intervals ought to be filled out with consistent sines, with the given intervals being cut into five equal parts, by this way.
Being taken then of these primary Differences which being called the Secondary Differences, and of the Secondary Differences the Third, and thus henceforth.
Let the given Sines and of these the First, Second, Third, etc, Differences in this manner being set out in distinct order.
2. If any Arcs permitted shall be equidistant; the Differences of the Sines of these Arcs are proportional to the Sines of the complements of the mean Arcs1.
For let the Arcs of 14, 28, 42, 56, 70, 84 Degrees be AB, AC, AD, AE, AF,AG. The differences of the Sines are HB, IC, KD, ME, NF, OG; If AB, the Chord of 14 degrees, shall be the total Sine; HB will be the sine of the angle BAH, 83, and CI the Sine of the Angle CBI, 69; DK, 55; EM, 41, FN, 27; GO, 13. Also HA will be the sine of 7 Degrees; IB, 21; KC, 35; MD, 49; NE, 63; OF, 77.
And because the First Differences shall be proportional to the Sines of the equidistant Arcs; the Second Differences for the same reason will be proportional to the Sines of the given Arcs which are the complements of the mean Arcs. By the same method of Differences, also the Third and Fourth and the rest.
The Second Order Differences, the Fourth, the Sixth, Eighth, etc are therefore proportional to the Sines given themselves. And the First Difference, the Third, the Fifth, the Seventh [are] proportional among themselves and to the Sines of the complements of the mean Arcs.
And from all of this so with the Sines as with the Differences, which have been placed in a like manner are proportional; the first and the third, the second and the fourth, correspond in this manner with the Degrees.
Over and above, all of these numbers situated in the same line shall be in continued proportion.
And since these being situated further up in the same line are in continued proportion. Truly these following as they are proportional are similarly placed, but [are] not in continued proportion.
And being given these Differences, and if the work will have been put to rights by the rule of proportion, by the above method expounded.
Now by these given other Differences have been sought, which serving for the insertion of the rest of the Sines. These being found either by division or multiplication of the given [ones].
If five shall Divide the First; 25 Divide the second; 125 the third; 625 the fourth, and so on with the Divisors being increased by a ratio of five; The first Quotients shall be the first means, of the second the second, etc.
Or if we should multiply the first Differences given by 2. The Seconds by 4. The Thirds by 8, etc., with the products increasing in the duplicate ratio, and we cut a single place and this the final of the product from the first, two places for the seconds, three for the thirds, etc; these products altogether being equal the Quotients by the Division first found; And these mean Differences in this manner should be placed.
And these are the mean Differences which have been corrected before we can make use of their help in order to find the Sines sought.
The method of correcting has been set out by me in Chapter 13 of the Arithmetica Logarithmica in the London Edition: but that chapter together with the following, without consulting me and being unaware, has been omitted in the Batavian [Dutch] Edition: but not in all things, the Author of this other Edition, in other respects a man industrious and not unlearned, seems to have understood my mind : Therefore lest anyone should fall short in any of this, who would wish to complete the whole of the tables, for something being judged of the greatest necessity ought to be transferred from there to this place2.
These mean differences are to be corrected by the following numbers placed in the following table: The First Column A of the same shows the mean Differences from the first as far as the twentieth. These Differences should be augmented with the correct Differences placed in column B being added and taken away these that have been placed in the following nearby column D. Hence by the same way being shown: by adding B, D, F, H and taking away C, E, G, I, if the number of different Differences shall increase.
The numbers placed in Column A are the mean Differences, of the first, second, third, etc, which have been corrected by the following numbers placed in the following Columns in this way3.
To the sixth mean, have been added 6 (8), namely by six of the eighth mean corrected: and besides being corrected by 26 (12) + 20 4 (16) + 4 08 (20). From the sum of these should be taken away 16 2 (10) + 27 6 (14) + 10 76 (18). With these correction Differences being added or taken in this way, what has been left will give the correct sixth Difference. Which everything in the following example will try to show.
4. With Sines the mean Differences should be increased and decreased, as we have shown. But with Tangents, Secants, Logarithms, and powers or figures with equally spaced roots [i.e. powers of integers and the like], the mean Differences being corrected by subtraction alone of all the correction Differences placed in the same line with the means.
For Sines, Tangents, Secants and Logarithms, and in everything with the ratio surely increasing or decreasing: If there shall be irrational [numbers]; it will be most suitable [to have] two places in the more removed of the Differences; but with the First and following at least one place beyond the limits to be conceded; as safe and sure (with one place always being added or taken, if required) we are able to find the correct differences and these numbers sought. As with the example of the Sine above placed on page 37 [here the previous page] and soon following . Where the fifth and fourth Differences have two places beyond the line which is the boundary of the given Sines and of the genuine Differences: But the 3rd, 2nd, and 1st Differences have as much as a single place beyond the same limit place.
The fifth and fourth Differences in this example not being corrected; because not having seventh and sixth Differences: but the third, second, and first being increased in this way.
See the Notes, where a similar table has been calculated, 12-9A.
5. Also with rational numbers, as with homogeneous powers of equidistant sides [i.e. powers of numbers such as the integers]; it will not be necessary to continue the corrected Differences beyond the given limits: because for all of these, as the powers sought shall be rational, so the Differences [whole numbers].
For let the cubes by the squares be given, or the sixth powers of the sides [numbers] 50:55:60:65:70: I desire to find the powers inserted between for the sides 51.52, etc.
Example of the Corrected Differences.
The cubes by the squares sought, together with these Differences actually found by the corrected mean differences.
1 These follow from the identity
sin(n + 1) - sin(n) = 2 sin(/2). cos((2n + 1)/2 = 2sin(/2). sin(p/2- (n + 1/2)).
Thus, e.g. if = 3.1250 and n = 3, then
sin(12.5) - sin(9.375) = 2 sin(1.5625) .sin(79.0625) = 0.053544141.
As 2 sin(1.5625) is constant for the set of differences, it follows that the differences are proportional to the sines of the complements: recall that the cosine had not been defined as such at this time. As Briggs indicates, the next set of differences revert to sines related to the original angles.
2 i.e. from the Arithmetica to the envisaged work. Thus Briggs did not acknowledge that Vlacq had in fact finished his tables, at least in the manner he had intended; and still held hopes that someone would do so; Vlacq was to upset matters again by not adopting Briggs' method of using decimal fractions of degrees rather than minutes and seconds, in subsequent publications of tables, after the death of Briggs. This relic from antiquity has since stayed with us. Thus, the 'golden moment' that Briggs wanted to seize had slipped away.
3 It is worthwhile to insert here the argument presented by Herman H. Goldstine in his book: A History of Numerical Analysis ... (Springer-Verlag, New York, 1977, pp. 27 - 30.), which formalises the numerical scheme used by Briggs for interpolation or sub-tabulation. It is evident that Briggs had discovered this process, though he would not have used an algebraic notation, but rather relied on the position of the number in his table to indicate what it was, or was the result of doing. So, following Goldstine's lead, who also provides references to the work of later mathematicians such as Lagrange and Legendre, subsequently dealing with interpolation in a way similar to Legendre.
Before doing this, let us summarise the initial state of affairs, as presented in the following Table 12-9A:
1. The sines of the angles of multiples of 25/8 = 31/8 degrees are given, having been found by the methods considered in the earlier chapters, though Briggs does not seem to indicate by which particular route he went to get the sine of the above angle.
2. The uncorrected differences of the various orders up to five have been evaluated for these angles, each one obtained from the previous by subtraction and division by 5. The odd orders are placed in the second slots, while the even orders are placed in the zero slots. To the accuracy considered, the 5th U. D.'s are constant. We show these in bold type
3. The sines of the sub-multiples of these angles on division by 5 are required to the same accuracy, formed from their corrected finite differences, by the process of sub-tabulation. Relations are found between the intervals of length 5 and those of unit length, for the subdivision, for the various orders of differences, which in addition are different for odd and even differences. This is explained below in the next note,
4. Briggs has observed that the higher order differences are zero, to this degree of accuracy: in this case the fifth order is observed from the table to be constant, and hence this order can be filled in immediately, and the fourth order thus differs by constant amounts, and can also be filled in. We have done this with italics. The remaining differences are to be corrected, according to Table 12-7, where some smart thinking has been done by Briggs. See note 4.
4 A unit shift operator E is defined for a function f(x), initially assumed to be increasing in the interval considered, which satisfies E f(x) = f(x + 1)
Now, in a numerical manner, Briggs defines the 1st order mean difference by the relation (f(x + 5) - f(x))/5, which corresponds to (E5 - I)f(x)/5 in operator notation, where I is the zero operator. A 1st order mean difference of this form with n = 5 intermediate steps, with the result placed in the second slot, can be written symbolically:
, or even as:
Now, it is useful to know that:;
and subsequently that: , where
(Note: etc). Hence, symbolically, the 2nd order difference from the 1st order difference that occupies the 0th point of subdivision, can be expressed as:
Hence for even powers 2p where p is an integer 1:
while for odd powers 2p + 1:
Similar results are derived by Goldstine, who refers to forward differences only, which is rather misleading, as Briggs uses both forward and backward differences to get his central difference results: though the same coefficients are obtained as in Briggs' Table 12-7. A great simplification is obtained if the differences in the final column are considered to be equal, which is the case if the results are being calculated to a finite degree of accuracy, by setting equal the different row levels in the final results for (12.10 and (12.2). In the case considered with constant 5th order differences, to the accuracy required, it follows that the 3rd order differences are also correct for a given row, and so the 1st order - from which it follows that the 2nd and 4th orders are also correct. There is hence a great deal of sense in using differences that obey these rules, in easing the arithmetical work. Let us see how this works out in practise:
When p =1, we have the corrected 1st order mean, obtained from the original mean, written in the second slot, together with the corrected 3rd order mean, and 1/5 of the 5th order mean; :
Now, this is the final result for the log function, or any other f(x) that increases monotonically, as we shall see with the 6th powers of integers tackled a little later in the Chapter as a sort of tour de force to vindicate the method; however, for the sine function, the finite differences of differing orders have signs attached - following the same rules as differentiation - as the various differences are themselves either increasing or decreasing functions in the interval considered: hence, the 3rd order differences are made negative:
to give agreement with Tables 12 - 7 & 8. To solve (12-1 & 2), the various levels can be expanded out to give Briggs' Table 12-7. Thus, for 2nd order:
Now, the initial 1st order difference and subsequent odd orders occupy the 2 mod(5) slots, while the initial even orders always occupy the 0 mod(5) slots, in order that the differences are centred on the place in the table being interpolated. The rest of Table 12-7 now follows:
5 An attempt has been made in Table 12-13A below to show how the interpolated values can be built up from these calculated, in bold, with the corrected mean written immediately below. The two columns to the right are not corrected, the 6th mean being 6!
There are a number of typographical errors in the original to Table 12-13, which have been corrected here.