The logarithms of intermediate numbers can be calculated in many ways. I consider embracing the present method especially; we shall see about the rest afterwards.

The first, second, third, fourth, fifth, etc. order differences of the given logarithms being taken; and the first order difference being divided by 5, the second by 25, the third by 125, etc ; on increasing the ratio of the division five-fold [successively]: the quotients being called first, second, third, etc, mean differences; or preferably in place of division, it may be done by multiplication of the first given difference by 2, of the second by 4, the third by 8, etc, cutting off one figure in products with the first order, two with the second, three with the third, etc [thus: instead of dividing by multiples of 5, multiply in turn by ²/₁₀ , ⁴/₁₀₀ , ⁸/₁₀₀₀ , ¹⁶/₁₀₀₀₀ , ³²/₁₀₀₀₀₀ ]. These products which are equal to those quotients, will be the mean differences of the first, second , third orders, etc. For let these logarithms [in Table 13-1] together with their differences be given for the first, second, third, fourth, and fifth orders, which are shown by calculation from the given logarithms themselves¹ :

For the required mean differences are nearby [Table 13-2], by multiplication of the first differences given by 2, with one figure taken away from the products. The rest becoming means if the numbers given are multiplied by 4, 8, 16, 32, [and the final 2, 3, 4, places dropped], etc.
These means are then to be corrected in this way.

For the two most removed: to wit the fourth and fifth order differences, do not need correcting (because the sixth and seventh [order differences] are zero: but for the other differences, the correction shall be by taking away of other more removed and correct differences: for by taking away the seventh has corrected the fifth : by the sixth the fourth, etc). The fourth and fifth mean orders may therefore be taken as the fourth and fifth correct orders .
So the third means are being corrected if from themselves are taken away three times the fifth mean differences.

From the second mean are taken away double the fourth correct mean, in addition should be taken away 1 ²/₅ of sixth order, if any of the sixth (order) have been found between these limits.[The result is the correct 2nd order mean difference].
From the first mean are taken away together the third correct mean and ¹/₅ of the fifth.
[The result is the correct first order mean difference].

And by this method all the differences have been corrected, and following from the [author's] own carefully prepared work. We would have been using the same method if there had been more different orders, by starting from the smallest and most removed [i.e. highest order]. As for how much from a given order should have been taken away, this table set out below indicates.

The numbers being placed in column A, designate the different [orders of] means, first, second, third, etc., as far as the twentieth. The numbers in columns B, C, and D etc., show the kind and quantity of the correct differences [that] have been taken away, from these different means in column A, which have been placed on the same line [row of the table].

Not all [correct means] have been taken away only for logarithms; but also for Tangents and Secants, etc., all the numbers at the same distance with the same power [i.e. they are either positive or negative]. But for the Sines, the differences confined to columns B, D, F, H have been added to the differences placed in column A: the rest namely in columns C, E, G, I have been taken away from the same.

The occasion for an example: from the sixth mean difference are taken away 6 [times] the correct eighth mean difference; 16 ²/₁₀ [times] the tenth; 26 [times] the twelfth, etc, by the same method [as] from the first mean difference, are taken away the [whole of the] correct third [mean] and ¹/₅ of the fifth. [As the previous table illustrates].

After those correct differences having been found, it is the most accurate, as each [figure] being conveniently placed in its own position, so that in the business of multiplication, every amount will be made able to avoid confusion. But this we can follow easier if we should have a sheet of paper and in this way with small open spaces separated by ruled lines, and if the first, third, fifth, seventh, etc. [columns] are being written with a colour different from the rest. The given logarithms are designated A, and by which the fifth place is occupied. And the second [correct] differences C, the fourth E, sixth, eighth, etc. being located on the same line [row] with the logarithms and towards the left. The [correct] mean differences, namely the first B, the third D, fifth, seventh, etc., occupy the places of that space. Finally empty spaces have been filled up by starting from the left. With the addition of the fourth differences with the third having been accomplished, with the additions of the thirds to the seconds; and thus successively. And while adding we can add or take away unity according as the circumstances shall require [i.e. the rounding off]. For with these irrational [numbers] it has sufficed to have the closest actual differences to them, as we are not able to find the true [numbers] with accuracy. The reason for this, I have said in the beginning of this Chapter, with the product of the first differences by two, the last place must be taken away; here nothing is being taken away. For with the first differences, and the rest, I have considered one adjoining place beyond the limits of the constituents: as all will turn out surer and more complete. I recommend the same [thing] to make [tables of ] Tangents, Secants, and Sines. But on the other hand, with the powers of equidistant numbers², where all differences are whole numbers, no less than with the given number itself; all being held between the constituent limits, in which for all of the various differences the number has been definite [i.e. whole], beyond that, if equal sides serve as differences, they will not be able to go further. As with squares there are two kinds [orders] of difference, in cubics three, in biquadratics four, etc. And the most removed differences are always equal to each other, and with the [final] equal products from the homogeneous differences of the sides, being calculated in a continuous product from the indices of the same power and all the smaller [powers]. For if the side difference be 1, the final differences will be, with squares 2, cubes 6, biquadratics 24, with fifth 120, sixth 720, seventh 5040, etc. Evidently from the continued product itself: from 1,2 : 2; from 1,2,3: 6; from 1,2,3,4:24; etc. But if the differences of the sides shall be 3 the furthest removed differences, of the squares will be 18; being made from the square 9 by 2; of the cubes 162, being made from the cube 27 by 6. Of the fourth power1944, being made from the bi-quadratic 81 by 24, etc.

As with all these powers, so with Logarithms, Tangents and Secants; it will be necessary to join together in an unbroken series some more numbers, without which we will not be able to reach the final differences. As with the given example, from the other part 2110 [to] 2105, [and] from the remainder 2130 [to] 2135. But with Sines, if the sines of three equidistant arcs were being given, all the differences or the most removed were being found by the proportional rule, if it were necessary. For the sine and differences of the second, fourth, sixth, and the eighth orders are in continued proportional, and the first, third, fifth , and the seventh [order] differences are likewise in continued proportional amongst themselves. And as the second order differences are between themselves proportional, for the Sines themselves for that place answer, and in the same way, the fourth, sixth, etc. (orders of differences); thus for the first [order] between itself and the third, the fifth, seventh, are in proportion with the complementary arc of the mean sine [i.e. the cosine].
But I am made aware of, from my further studies, the increased progress of [my understanding of] the common origins as the laws [governing the formation of successive differences] are becoming apparent.

If to the same Chiliad you can suppose another of these being added, consider say the twenty first [Chiliad], the fifth part should be taken of that initial number that you are about to take: the first number will be 20000, of which the fifth part [is] 4000, to the logarithm of this, and to the nearest 200, one by one the logarithm of five being added, the sums will be the logarithms of five and of that number, through the whole of the same Chiliad: namely 20000, 20005, 200010, 200015, etc., and of which the first [order] differences are those themselves in the fifth Chiliad, being found within those two hundred logarithms [i.e. the Chiliad consists of 200 logarithms set out in multiples of 5]. From which the second [order] differences should be sought. Also, the second order differences will give the third order. Truly the fourth [order] are very small, which we will be able to ignore with safety, for that reason. Then the first [order] difference is multiplied by two, the second [order] difference by four, the third [order] difference by eight. The products will be the mean differences, which should be found from their own positions, cutting off by one place from the second, by two from the third [i.e. leaving the extra place for rounding off purposes]. Thus the first [correct order] have been fetched from those in position, then by subtraction the third [order] will have been corrected. All the rest have been completed by addition.

And this method of inserting four logarithms between the two given nearby, will be called quinquesection: as from each interval there becomes five. Also, there will be general laws being propounded for trisection and septisection. But of all these quinquesection is the most outstanding, we should look at either the convenience or ease of use. Nevertheless few will find it irksome to propound the trisection method . The first, second, third, etc. of the given differences being taken at first. Then, let the first be divided by 3; the second by 9; the third by 27; the fourth by 81; etc, by increasing the ratio of the divisor by 3: the quotients will be the first , second, third, fourth, etc., mean differences. These mean differences, as before should become smaller with everything, except with the Sines; then when corrections will have been made, they have among themselves been transferring places: and by starting from the smallest and furthest away, all are as before the addition being carried out. But how much and which kind of difference should be taken away may be, is indicated by the table at hand.

A First mean difference, being taken away is a third of the Third correction.
A Fourth mean difference, being taken away are ⁴/₃ of the sixth, ²/₃ of the eighth, ⁴/₂₇ of the tenth, and ¹/₈₁ of the twelfth corrections.

The remaining sections for which the names of the fractional numbers being assigned, as bisection, quadrisection, etc., are more difficult. Because even when we attempt to find the sub-tangent for the circle: with sections from unequal numbers in the denominator, themselves the sought sub-tangent, may be shown with one operation. The remainder namely with equal denominators, not themselves subtangents, but to such an extent they are expedient square for subtangents³.

¹Note the use of intermediate row levels to indicate the differences. The sophisticated method of subtabulation being propounded by Briggs in this Chapter is explained at some length in the Appendix.

² We would call this the base number of a sequence of powers. Thus, in the products nr, where the index r = 2, 3, 4, etc., and n has a constant increment, the final differences are equal.

By 'root' or 'side' Briggs usually is referring to the base number that had been raised to some power: here he means the result of the previous difference operation, presumably in analogy with taking a square root.

³The interested reader would do best to examine the Appendix to Chapter 13, produced as a separate document, where a modern approach is used to produce Briggs' correction coefficients. To fully understand the references to Sines, Tangents, and Secants, the reader should consult Goldstine or Hutton. This writer is at present engaged in translating Briggs' Trigonometria Britannica, when all should become apparent finally. Briggs' frustrations are partially due to the fact that his subdivision method using extended central differences only work for odd points subdivision, as the Appendix shows. Secondly, the idea of higher orders disappearing does not really work very well for periodic functions!