We present initially a small historical note of some interest, before turning to the problem of interpreting Briggs ' new interpolation scheme. Briggs has run into trouble interpolating logarithms using his method of forward finite differences, referred to in the last chapter. We recall that this method works well as long as 1st and 2nd order differences only are involved, which corresponds to numbers in the locality of a power of ten, when 14 decimal places are required. Thus, Briggs was able to find the first 20 Chiliades, and the 90 to 100 Chiliades using this easy form of interpolation. The scheme proposed here by Briggs to extend the log tables by subtabulation between known logarithms to cover the missing Chiliades was never implemented by him, although in his advice to the reader he was keen that someone should undertake this task.

Adrian Vlacq, a Dutch bookseller and publisher, who was to finance and oversee the first complete set of Briggs' logarithms for the first hundred Chiliades, got round the computational difficulties with the help of his associate, the mathematically inclined Ezechiel de Decker, by reducing the accuracy of the logarithms from 14 to 10 places, in which case the previous scheme of Briggs could still be used. Furthermore, Vlacq's 2^nd edition of Briggs' book omitted completely Briggs' original Chapters 12 and 13, [Thus, in Vlacq's edition, chapter 12 corresponds to Briggs' original Chapter 14,] a fact that Briggs was to bemoan in his subsequent work, Trigonometria Britannica, with just cause, for this is the main jewel in Briggs' crown -- as we shall see shortly. Also, material was moved around inappropriately by Vlacq in the new edition -- thus, the section on compound interest, being an application, was presented at the end of Chapter 7, long before the main development was complete; while most of Briggs' occasional typographical errors, were copied holus bolus.

Thus, as people would say now, Briggs' work was produced in a pirate edition purely for financial gain; on the other hand, Briggs had left the door open for such a development. He felt that perhaps his time would be better spent producing the logarithms of the trigonometric functions then in use. It is sad to note that the great structure of numerical mathematics produced here in this chapter has been allowed to pass into oblivion without comment: Hutton, in his commentary, has the gist of the mechanics of the method used for the interpolation, which is a straightforward matter, but remains silent on the underlying theory used in its production. This is partially Briggs' own fault, for he was secretive and revealed little: yet it is not too hard to uncover. The method was rediscovered by Cotes a century later: A note by D.T. Whiteside in Vol. 4 of Newton's Mathematical Papers, pages 43 -- 45 gives the history. It is hoped that the present rendering will help to do justice finally to this neglected work, and to present the outcome in an elementary way.

As the reader may have gathered already, Briggs does not appear to have reviewed the textual aspects of his work particularly well at the printing stage, perhaps due to no fault of his own. For example, there are two pages 17, chapter 19 is called chapter 15 [ there may well have been an insertion of extra material around this stage of the printing]; he had a habit of not labelling the first section in a chapter, and calling the following B, or 2; and often did not label new sections at all, but moved as the spirit takes him on to a related topic, even in mid-paragraph at times. Like many another author who understands his work very well, he would jump into the middle of an explanation at the start of the chapter.

These minor criticisms pale into insignificance when one considers the organisation of mind over matter needed to bring a task of this magnitude to a successful conclusion. For Henry Briggs acted as his own computer: he had to think out what calculations to perform, how to effect them with the minimal effort, and insure their correctness, file them away in an orderly retrievable manner, and eventually bring everything together as a whole for publication. As this present work is meant to be a translation, there is no justification to alter Briggs' exposition: though small amounts of material inserted into square brackets [thus] is generally put there to make the arguments flow more easily. Needless to say, the writers who followed Briggs were more concerned about making the tables easier to use, and less with their construction: eventually through their efforts, the user could stay blissfully ignorant of Briggs' trials and tribulations, and of his eventual triumph, in order to produce these wonderful tables: for there was by then no need to understand their inner workings -- though the same tables were to prove to be the fundamental computational component in all kinds of applications, wherever tedious arithmetical calculations arose, endlessly over the next 350 years, until the advent of modern computers.

[Anyone not fully convinced of this fact should read, perhaps, the autobiography of Neville Shute, aptly called Slide Rule, concerned in part with the design of airships in Britain in the 1930's.] Nowadays, an unaware person looking at the extent of Briggs' original tables could be forgiven for thinking they too had been produced by a computer of the electronic variety.

Let us now look in some depth at the amended interpolation formulae proposed by Briggs, who was greatly hampered by the lack of a suitable notation at the time. We will attempt to correct this deficiency: however, the reader should be aware that none of this material is present in Chapter 13 of the Arithmetica, apart from the table of coefficients which we call 13-3.

Briggs did not give an account of the procedure he used for producing these coefficients, but we surmise that is must have been close to the one presented here. The intention being as follows: given a set of numbers and their logarithms, these being taken as multiples of 5 as Briggs does in his initial dissection of the interval between numbers with known logarithms -- to find the logarithms of the 4 intervening equally spaced numbers between consecutive members of the table, to the same degree of accuracy.

Thus, in Briggs' example Table [13-1], the numbers shown with known logarithms are 2115, 2120, and 2125, being part of a larger set of such numbers. We are required, by using some kind of finite difference method, to find the logarithms of 2116, 2117, 2118, 2119, 2121,É etc -- a process known as subtabulation, which is described in older texts on numerical analysis -- as the process is now out of use, with the advent of electronic computers. Table [13-1] is thus expanded to Table [13-4], and we can think of the larger table being present initially with numerous blank spaces, the task being to fill them in according to Briggs' method, and to reconcile the results with our present day understanding. The former is more or less the task that Briggs had set himself, and he recommended leaving blank spaces, using inks of differing colours to avoid confusion, and the like.

As with a lot of Briggs' work, we can expect a 'non-standard' procedure to be used, made even more difficult to unravel by the lack of any explanation offered. It is well-established that Briggs' contemporary Harriot, the other prominent English mathematician at the time, had investigated subtabulation in a fairly sophisticated manner, and it has been conjectured by Goldstine that Briggs had acquired these interpolation techniques from Harriot. [A History of Numerical AnalysisÉ, H. Goldstine Springer,1977; p.26].

However, Briggs is very careful to acknowledge the assistance of others, either dead or alive at the time, in his work, such as Viète and Gunther. Thus, though it is probably true to state that Briggs knew about Harriot's work in this area, it is also probably the case that his own methods were peculiarly his own, such as those we shall discuss, or were considered common knowledge at the time, as with those in Chapter 12. We now proceed to follow Briggs' explanations of the mechanics of the method, but insert out own notation.

There are three kinds of finite difference introduced by Briggs:

(i) The 'raw' differences in Table 13-1, that were called simply a, b, c, d, e, etc. in the previous chapter (See equations 12.1 and 12.2). Now these differences must be incorporated into a larger scheme, and a more detailed notation is required. The rows are related to the absolute numbers (as the argument of the log function is called by Briggs), and we know the logarithms of those which are multiples of 5; while the columns refer to the order of the difference. The symbol _5jⁱ will be used to denote the i^th forward difference 'in line with' the 5j row, where j is a row index that can be set in an arbitrary fashion. Thus, i = 0 is the zero^th difference, and corresponds to the logarithm of the number at this level: We take the logarithm of 2120 in Table 13-1 as ₀⁰, while _-5⁰ and ₅⁰ correspond to the logarithms of 2115 and 2125 respectively, all in column two of the table.
The index i = 1 gives the first order difference; Initially we note that these undivided first order forward differences have the form

_5j+2⁰ = _5j+5⁰ - _5j⁰         [13.1],

_5j+2⁰ = _5j+2⁰ - _5j+2⁰, _5j+2⁰ = _5j+2⁰ - _5j+2⁰ , etc.          [13.2].

_5j+2¹ = (_5j+5⁰ -_5j⁰)/5          (B);
_5j+2² = (_5j+2¹ -_5j-3¹)/5          (C);
_5j+2³ = (_5j+5² -_5j²)/5           (D) ;
_5j⁴ = (_5j+2³ -_5j-3³)/5          (E); and
_5j+2⁵ = (_5j+5⁴ -_5j-3⁴)/5           (not in Table 13-1)          [13.3];

(iii) Briggs' Correct Differences (C.D.) is denoted by _j⁽ⁱ⁾. The index (i) = 0 thus applies to the logs themselves, and we have in Table 13-4:

_5j⁽⁰⁾ = log 2120; _5j+5⁽⁰⁾ = log 2125; _5j-5⁽⁰⁾ = log 2115, where j = 524.

According to Briggs' example, the corrections required are as follows (i.e. to convert the various orders of differences into useful differences that can be used to insert the values of the logarithms into the empty slots in the table with steadily increasing accuracy):
For the 5^th and 4^th orders, no change needed, as higher orders are negligible at the required precision for the example provided.
Hence, _5j+2⁽⁵⁾ = _5j⁵ and _5j+2⁽⁴⁾ = _5j⁴;
3^rd order: no change needed here, as the 5^th order is zero in this part of the table to this precision;
Otherwise: _5j+2⁽³⁾ = _5j+2³ - 3._5j+2⁽⁵⁾; (See Table 13-3).
However, for 2nd order: _5j+2⁽²⁾ = _5j+2² - 2._5j+2⁽⁴⁾ - 1.4_5j+2⁽⁶⁾ = _5j+2² - 2._5j+2⁽⁴⁾ in this case.
For the 1^st order: _5j⁽¹⁾ = _5j¹ - _5j⁽³⁾ - 0.2_5j⁽⁵⁾ = _5j+2¹ - _5j⁽³⁾          [13.4].
Perusal of Table [13-4] shows how these means occupy various levels called A, B, C, D, and E for orders 0, 1, 2, 3, 4, and 5 respectively. Noting as above that the 1^st (B) and 3^rd (D) are given the (5j + 2)^th intermediate position, while the 0^th (A), 2^nd (C), and 4^th (E) occupy the 5j position, and these are used in an iterative scheme to produce the intervening logarithms, as we show here :

_j'-2⁽²⁾+ _j'-1⁽²⁾ = _j'-1⁽²⁾, _j'-1⁽¹⁾+ _j'-1⁽²⁾ = _j'⁽¹⁾, _j'⁽⁰⁾+ _j'⁽¹⁾ = _j'+1⁽⁰⁾ [13.5]

It is observed that the above relations are satisfied to this degree of precision in the Table 13-4A, as this example shows. This completes the mechanics of Briggs' method, for the simple case he has presented.

1. Our first task is to write down the various differences in terms of the known logarithms of numbers in the first column in the right-hand side of Table 13-1. We will use this convention: the function notation log(x) is used for the variable x, but otherwise we omit the brackets if the logarithm of a particular number is being evaluating.
For a given j, we have:
Zero^th Order:

(These are not differences of course, but are included here for consistency with the notation):
First Order Differences:

Second Order Differences:

Third Order Differences:

Fourth Order Difference:

Fifth Order Differences:

The next task is to check that the expressions developed for the differences in terms of the logarithms of known numbers, equations [13.6] to [13.10], give the correct values in Briggs' Table 13-4; this is done using a spreadsheet:

This table illustrates that Briggs' uncorrected differences are reproduced by the above difference formulae, as expected, by comparing with Table 13-4. The odd differences are placed in the '5j + 2' slots, while the even differences are placed in the '5j 'slots.

We see that a pattern has emerged for the odd and even differences. It is convenient for further development to introduce the operator notation that is so useful. In particular, for a function f(x), the unit displacement operators E and E ^-1 are defined by Ef(x) = f(x + 1) and E ^-1f(x) = f(x - 1); while E⁰ = I, where If(x) = f(x), the identity operator. In our case, f(x) is log(x), and the 1st difference [13.7] can be written in the form:

The lower left-hand index 5 allows four intermediate points of subdivision and 5 equal intervals (we will suppress this index until it is required later for a value other than 5), the lower right-hand index gives the point where the value of the function is approximated, while the upper index gives the order of the difference. The process of subtabulation is best described using this method: the resulting operator can be applied to any position in the original table. We will assume it acts on the 5j position. Note that successive differences are always defined relative to their initial starting off point, even though they are placed at some subsequent level in the table: e.g. the 2nd order difference for the number 2120 lies at the level of 2125, but has been generated by j = 424. (This may be a source of confusion). The intention is to correct all orders derived from the initial table of logarithms, where the spacing is 5, to the correct differences for single spacing, thus allowing the subtabulation to proceed. This is done by using Briggs' 3rd table, which shows how to correct given differences in terms of already correct differences, starting from the 20th at the top of the table. We will show how the coefficients can be determined.
Succeeding differences are defined as follows,:
2nd Order:

3rd Order:

Hence for all orders p > 1, the operator equation for subtabulation into 5 holds:

These results, or similar ones, are established by Goldstine, but the origins of the method are much older. Although the operator method is wonderfully effective, it needs to be applied in order to get a better understanding of how it works. We look at the last form of [13.14] in more detail:

The operators act on the original logs: and we choose log(5j) to illustrate the proceedings, taking this number as the zero index:

The higher orders mean differences that we require are similarly defined:

The correct differences are similar in form, but act between adjacent whole numbers, rather than multiples of 5:

Now consider

hence: , as required. This is the method used by Briggs to change from intervals of 5 to unit intervals for the 1st difference of logarithms.

Table [13-3] sets out the coefficients which follows by iteration of this result.

Thus,

giving:
, etc, for higher orders up to the 20^th. The coefficients are those in Briggs' Table that we call [13-3].

A Mercator expansion of an nth order difference, either corrected or not, has the first n - 1 powers zero, and the same first non-zero term. Hence, in any table where this method is used, the highest order difference that is non-zero does not need to be corrected, and the correction terms required can be evaluated by reading the appropriate coefficients from Table [13-3], proceeding to lower orders: which is done by Briggs in his example.
It is then a fairly trivial exercise to see how the correct differences actually work, which depend on the properties of logarithms. As the order of the correct difference increases, the argument tends to one, and its logarithm to zero. Eventually only the first term in the expansion is needed, where the difference assumes a small almost constant value, at the required accuracy. We choose to work backwards for convenience, to justify the formulae in [13.5]:

At the end of the chapter, Briggs considers other subdivisions. In particular, for trisection, we can show how the table of coefficients we have called [13-5]:
First Order:

Second Order:

3rd Order:

Hence: , for p =1, 2, 3, etc. By iteration the table is completed.