If the second order differences of the logarithms of the given numbers are almost equal, it will not be difficult to show this: otherwise, if third order differences have been used, this method still will not be found lacking.

Two nearby numbers A being taken, and their logarithms B, together with the first and second order differences of these, C and D. [Table 12-1.] If the second differences are equal, the second of these (D) being multiplied by the numbers in the Table E, [12-2], with the ten numbers being added on to the first. Now the numbers F, G, H, I, K having been worked out [Table 12-3], with the last three places taken away: for the first five being added, with just as many being taken away from a tenth part of the first difference C, which is added on in each case¹. The totals and the remainders are the differences of the logarithms we seek; which for the given smaller logarithm, by itself with ordinary continual addition, will give the logarithms sought, as we see here. Thus, with the first numbers given 91235, 91236, and the first difference [C] 476014799:

If the second order differences are not equal as here the two numbers in proximity being added and taking half the sum for the second difference, and being multiplied as before.

If you want to find any of these, with the rest omitted²: a number less than ten is multiplied by the given difference C, connected with the given number A; and with the same corresponding number from the table E, let it multiply the second difference, and by removing three places, the products being added, with rounding and division by 10: the whole being added to the given logarithm B , giving the required logarithm. Thus if I wish to know the logarithm of the number 96157. The given difference 4516608416 [between 9615 and 9616], being multiplied by 7, making 31616258912. Then 105 located in Table E from the seventh position, let it multiply the second difference 469721, making 49320|705 with the three final places taken away, being added to the first product, from which one decimal place has been removed³, 3161625891, total 3161675212 being added to the given logarithm . The total 3,98298,09053,2662 will be the logarithm of the number sought 96157 **.

⁴If you wish to know the difference between the logarithms of this number and the next larger: the number 8 in the Table corresponding to G, which exceeds all of the seventh part, let it multiply the second difference 469721; the product 11743|025 with three places subtracted, being taken from the tenth part of the given first difference [H], the remainder 451649099 will be the difference sought .

And so by this method the proportional part may be found with sufficient accuracy. For following [the methods] which were being considered in the above chapter [11], we will be transferring the given logarithm to the appropriate place in the ultimate or the penultimate Chiliad, [where]we will be able with the first place of the proportional part, to add to the absolute number found in the Chiliad, and the numbers being increased to discover the logarithm through those nearest preceding, together with the difference between the same of that and close to the larger. This difference will give the proportional part of that thing which we seek most accurately, as we may see by a single example.

Being sought is the square root of 147. Given the logarithm [of 147 is] 2,16731,73347,4814, of which half 1,08365,86673,7409 is the logarithm of the root being sought.

Hitherto by the previous notions of Chapter 11, that gave the place as 7, with the number 99322 in the table found nearby. The remainder required following that preceding [i.e. 993227]

Therefore the number corresponding to the logarithm NN 0,99704,86110,0584 is 9932272150923.

Thus if the number 8192, or the factors 8.8.8.8.2 of the same, divide the number found, the quotient 12124,35565,29831,6 will be the root of the number 147 required and enlarged. It is thus close to the true root, given by⁵ 12124,35565,29824,41054.

¹ This is an appropriate place to quote Charles Hutton, from that uniquely wide-ranging and illuminating preamble to his Hutton's Mathematical Tables 5^th Ed., (1811): on the Construction of Logarithms, page 71:

... our ingenious author first of all teaches the rules of the Difference Method, in constructing logarithms by interpolation from differences. This is the same method which has since been largely treated by later authors, and in particular by the ingenious Mr. Cotes, in his Canontechnia. How Mr. Briggs came by it does not well appear, as he only delivers the rules, without laying down the principles or investigation of them. He delivers the method into two cases, namely when the second differences are equal or nearly equal, and when the differences run out to any length whatever. The former of these ... he particularly adapts to interpolating 9 equidistant means between the two terms, evidently for this reason, that then the power of ten becomes the principal multipliers or divisors, and so the operation performed mentally. The substance of his method is this: having given two absolute numbers with their logarithms, to find the logarithms of 9 arithmetical means between the given numbers: Between the given logarithms take the 1st difference, as well as between each of them and their next or equidistant greater and less logarithm: and likewise the 2^nd differences, or the two differences of these first three differences; then if these two differences be equal, multiply one of them severally by the numbers 45, 35, etc, as in the annexed table, dividing each product by 1000, that is cutting off the last three figures from each; lastly, to ¹/₁₀ ^th of the 1st difference of the given logarithms to add severally the first five quotients, and subtract the other five, so shall the ten results be the respective first differences to be continually added to compose the required series of logarithms. Now this amounts to the same thing as what is at this day taught in the like case: It is known that if A be any term of an equidistant series of terms, and a, b, c, etc, the 1^st, 2^nd, 3^rd, etc order of differences; then the term z, whose distance from A is expressed by x, will be thus:

z = A + xa + ^x(x-1)(x-2)c/₆ + etc.      [12-1].

And if now, with our author, we make the 2^nd differences equal, then c, d, e, etc, will all vanish, or be equal to 0, and z will become barely

= A + xa + ^x(x-1)b/₂      [12-2].

Therefore if we take x successively to be ⁰/₁₀ , ¹/₁₀ , ²/₁₀ , ³/₁₀ etc, we shall have the annexed series of terms with their differences. Where it is to be observed, that our author has reduced the differences from the 1^st to the 2^nd form, as he thought it easier to multiply by 5 than to divide by 2. Also all the last terms ^x(x-1)b/₂ are set down positive, because in the logarithms b is negative.

² i.e. the last place, after dividing by 10.

³ Inspection of Table 12-5 shows that the 2^nd order sums of products are formed with an extra place, which is then rounded.
Note that the differences are added cumulatively in table E, and not added/subtracted to the previous difference, as in Table 12-3.

⁴ This follows in a straightforward manner from 12-1 and 12-2.

⁵ Thus, the method of the previous chapter is used to obtain an approximate value for 147 by first finding the number closest to a power of ten in the upper Chiliades by considering their logs: Briggs finds 147 8129 = NN has a log that lies between 99322 and 99323. Simple proportion then gives this number more accurately to be 99322.7. However, with the power of the finite difference method available, and as the second order is sufficient as we are so close to a power of ten, the logarithms corresponding to 99322.7 and 99322.8 can be found to the full 14 figure accuracy, between which the log of NN is located. Hence, by simple proportion (again) the accuracy of the NN is increased dramatically, as Briggs shows, and so with the final value for 147.