In Chapter 11 we showed how a number that was being found between the Chiliads, it was possible to be increased through proportional parts, so that the place of the error should be scarcely within the twelfth place. Now I will show in the same way with help from the nearby preceding table [i.e. Table 13-7.] through only subtraction and multiplication, namely it is possible to find the number being sought: if the number by the method is rational, it will be able to be described to fourteen places, and [otherwise] with fewer places.

The numbers which being confined to this table, being set out in nines, which are on the left, unity, and with the ones some other small fractions. Their logarithms have been placed by the right exactly opposite. The first nine are the single unit numbers, being designated by the letter A, with the next nine being designated by the letter B, are one and the decimal fractions of one; thus, 11, 12, 13, etc. The next nine being designated by the letter C, are one and the hundredths, with the rest of the of nines successively in like manner. D thousandths adjoining with unity, E tens of thousandths, etc. With the aid of these, in this way the absolute number being conveniently found for the given logarithm. Let the given logarithm be 3,66067,57883,3852. From this (where the characteristic will have been removed) the third logarithm 0,47712,12547,19662 having been selected from the first of nine being taken away, there will be left 0,18355,45336,18858: from which the logarithm of 14 can be taken away, that this table shows in the second of nine, then from the next of nine 109 has been taken away, and there will remain nothing. I assert the continued product from these three 3, 14, 109, 4578 to be indeed the number being sought. For the whole operation takes account of itself thus:

But the most convenient way will be to take from the Chiliads the next smallest logarithm given, which being taken away from that given, and of which the absolute number found being noted in the margin. The rest indeed should be carried out with the help of the preceding table. For let the given logarithm be 3,48314,00744,3475 the nearest logarithm in the fourth Chiliad was found, 3,48301,64201,4413 which being taken from the given left 0,00012,36542,9082. By noting in the margin 3041 found, by seeking the remaining logarithm from this table, or of that nearest: that by being subtracted, noting down the same absolute number, and I progress thus as before. Here you have this way of working described.

In this operation we have twelve logarithms, the first of which being taken from the Chiliads, the next six one by one from the single nines of this table, to be leaving namely five from the final of nine. As for subsequent nines and the last, so much of this lay between; because the subsequent significant places have more ciphers in front; we can find the position of that final (factor and logarithm) by a not inconvenient subsequent exercise; if we were progressing further with that, we should truncate more of these places; because we will have been able to observe the factor in what has gone before. These twelve logarithms, being collected in one sum, being equal to the given logarithm: but the numbers being found from the opposite side, are factors, which by continued multiplication produce the number agreeing with the given logarithm. Whence by multiplication a little change may begin to be produced, by way of the product of the first by the second, being multiplied by the third; and the following product by the fourth: and successively with these in the same way, until with single factors being left, the preceding products will have been multiplied together. As 30401 by 10002, the product will be 30416082, which we multiply by the third 100008, makes 3041851528656, as you see here:

Finally, these factors of ours gave the number being sought 304186597056004.

Above we have shown, how for a given logarithm, the agreeing absolute number could be found; by subtraction of the logarithms in the preceding table together, and continued multiplication of the numbers from their directly opposite position: here in contrast being shown, how for a given number the agreeing logarithm may be sought; by division of the given number, and by the addition of the matching logarithms of the quotients. If the number given being written with as many as four places, the logarithm of this [number] having being described in the Chiliads. But if with more [places]; the first four places of the number being taken together (except on account of a strong will, by trying with another method, all shall turn out the same eventually, with two or three more places might be pleasing). You may thus divide this given number, as after this division you will have taken away half from the dividend, the product from the divisor taken with the quotient found, always being added to the divisor, but the logarithm of the particular quotient being found in the table, being added to the divisors of the first logarithm: the total will be the logarithm of the given number sought. Let the given number be 3041851528656, I divide this number by 3041, which I subtract from the number being divided, noting down the quotient 1 : 851528656 will be left. Then I continue, and after three ciphers I place 2 with the quotient, and the product of 2 [first product] with the given divisor I subtract, writing below the remainder. Then the same quantity I add to the divisor [ i.e. the extra digits 28656], and increase [i.e. 243300000 + 28656] by the same, I divide the remainder; the product formed being taken away from the dividend , and adding the same divisor [i.e. an 8] to the quotient; then as before. Here you will be able to comprehend the method of the whole operation.

For all of this that has been demonstrated has been taken from the second chapter. For there have been given a divisor A, and a second quotient B, with the factors of the divisor being increased C: which other [number] has been given to the divisor, and without harm will be able to be called the second divisor. And of these the logarithm being equal to the logarithms of A and B, by the axiom, Ch.2. Then the second divisor will begin the same work, dividing the whole of the given number, and finding the third quotient 10008. For only the final place of the quotient is permitted to be multiplied, and the product from the same only, being taken away from the remainder, so we know even the factor being taken away from the unit place with this division, which has preceded the discovery of these final quotients: therefore if the logarithm of this final quotient D, being added to the logarithm of the second divisor, the total will be the logarithm of the divisor again augmented E: it is itself that of the divisor.

So progressing with this division, and increasing the divisor of these, then with the increase of the factor until it will be equal itself with the given dividend, of which the logarithm is being sought. Which I will try to make clear with another example.

Let the given number of which the logarithm is sought be 296682051456, this I divide by 2966. All the sequence of the working you see here.

To this point [we have been concerned about] the making and of [our] disposition towards logarithms. A look at some of their uses follows [in the following chapters].