With numbers from unity in continued proportion, such as 1.3.9.27.243.729.2187., that number which is nearest unity, is said to be the root of every succeeding [number]. ['latus', meaning 'side' from geometry is used by Briggs here in various forms both here and in succeeding chapters: 'root' seems the most appropriate term for us to use]. The following [numbers] being said to be powers of the same root: because they become [so] by multiplication of the same root with itself, and among products from themselves. The second of those from unity being called the square, the third the cube, the fourth the biquadratic: of the rest still further away, the names being assigned, following that which they hold from the distance from unity. So the fifth is said to be the power five; and the rest in the same way. But yet the characteristics, of which these being recognised and distinguished in turn, are of numbers written within circles [actually brackets], as you see here.

In that 3 is the root of the square 9, the same 3 is the cube root of 27, and of the biquadratic 81, and the same root to the seventh power (which is the final) 2187. Therefore, following what was being propounded [in] the above chapter, the differences of the logarithms of the given numbers and of unity being taken, (that is with the logarithm of unity, 0 itself being the logarithm of a given number) and by the number of the interval between the given number and unity being divided; the quotient will be the logarithm of the root being sought. For if of the square the square root is being sought, the logarithm of the given root being half (the logarithm) of the given number; if of the cube, a third; if of the power (7), the seventh part, etc. This fraction will be the logarithm for the roots being sought. Let the given number be 15625 the sixth power. Of which the logarithm is 4,19382,00260. The (6) root is being sought. Being taken, therefore the sixth part of the logarithm found 0,69897,00043, of which the matching number is 5. I assert that 5 is the root being sought. As it appears along with these numbers, 1.5.25.125.625.3125.15625.


Let the given cube be 979: the cube root is being sought. The log. of the given number is 2,99078,26918,0314 of which the third of the total should be taken: 0,99692,75639,3438. Of this example, as with the above, the characteristic (as in Ch.4 we accepted to be the first place to the left) must be carefully respected. And so, as the third part of the first place given cannot be taken, put 0 in the first place of the quotient, for the characteristic of the same, which shows the root sought to be less than ten.

And as it is not possible to find the root with integers, we may think of the root being sought multiplied by 10000, of which the logarithm 4,00000,0000,0000 being added of the third being found, the total will be 4,99692,7563934538, of which in the final Chiliad corresponds 99295 and more. Which through differences and the proportional part, 99295,04202,067 being found to be accurate enough. It is 9929504202067, when nine places have been added on. Because now the two final places are (making the root) greater than it should, 067: for which 047 should be substituted, as I cautioned in Chapter 11, and showed in Chapter 12, how near to the true root sought we be able to come.

And by this way it will be able to be computing the root of any proposed power, if not exactly, at least approximately. And not to such an extent the root, but any other of the same series; either between those given roots and unity, or in steps further away. For let the sixth power given be 16525: the logarithm of the root has been found 0,69897,00043,3602. If I want to know the fourth power of the same root, by multiplying the logarithm of the root found by the number four, it is 2,79588,00173,4408, of which 625 is the corresponding number. But if I wish to know the ninth power of the same, by multiplying by 9, the product 6,29073,00390,2418 will be the logarithm of the same, the characteristic 6 of which has been shown, with the number being sought to be 7 places; and therefore to be bigger than it may be possible to find among these chiliads [recall that the numbers corresponding to Briggs' logarithms have an upper limit of 100,000]. Therefore, by reducing that first place, for 6 substituting 4, and it will be the log. 4,29073,00390,2418. Looking in the twentieth Chiliad, from which 19531 was the corresponding number. But as I wish to fill up the places, because this number being deficient; and the proportional parts being added from that Chiliad in fact may not be perfect; by the precept of Chapter 11, I take the complement of the given logarithm 0,70926,99609,7582 , because it was being sought among the logarithms on page 23 (just as with the rule and example shown on p.21) [p.11-2 in this work] where the number itself beyond hope is to be found, (and placed opposite 512) this being added with that given, gives 1,00000,00000,0000, to which the number corresponds completely (if we increase the characteristic) 100000. Which divided by 8.8.8, the factors of the number 512, gives the final quotient 1953125. As you see here.

This final quotient is the ninth power of the root 5 discovered previously.

The roots are also present with fractions, square, cube, etc., as you see here. And these roots will also be found through logarithms. For the defective logarithm being taken of the given fraction, [i.e. a logarithm less than zero], as we showed in Chapter 10. For the half of this will be the logarithm of the square root, a third the cube (root), etc. Let the fraction ⁸/₂₇ be given of which the logarithm will be
-0,52827,37771,6705 the cube root of which fraction being sought. A third of the given logarithm should be taken: - 0,17609,12590,5568. The number corresponding to this logarithm will be the denominator of the fraction, of which the numerator will be unity. The denominator will be 15. As it is a number less than 10, the characteristic will be 0. The root being sought is therefore ¹⁰/₁₅. If the square root of the same fraction is sought, half of the same logarithm should be taken - 0,26413,68885,8352, to which 18371173 corresponds the denominator of the root sought, of which the numerator is one. But because the numerator of the fraction should be 1, for this reason it can be shown, because the difference of the logarithms of the numerator and the denominator is the logarithm of the fraction, as we demonstrated in Chapter 10. Therefore if the denominator is being sought for the corresponding logarithm, it is unavoidable that the logarithm of the numerator shall be 0: that is, the numerator shall be 1. It cannot become otherwise, as [it is] being taken care of by the difference of those given logarithms. Because if we wish to express the same given fraction [parts] with other numbers, any number to be taken for the numerator, or for the denominator. If the numerator has been assumed, of which the logarithm* being added to the logarithm of the fraction: the total will be the logarithm of the denominator. As with the cube root of the fraction ⁸/₂₇ the logarithm has been found -0,17609,12590,5568. Let the numerator of the root being sought be 12, the logarithm of which 1,07918,12460, with that other being added gives 1,25527,25051, to which 18 corresponds, which will be the denominator. So the root being sought will be ¹²/₁₈. But if the denominator has been assumed, the logarithm of the fraction has been taken away from the logarithm of the given denominator, and the remainder will be the logarithm of the numerator. For let the denominator of the same cubic root being sought be 36, for which the log. 1,55630,25007,6729, from which the logarithm found of the root sought being taken away, 1,38021,12417,1161 will be left, the logarithm of the numerator 24, I assert the root sought to be ²⁴/₃₆ . For the fractions, being written one way or another, are good for the same value, if it shall be the same ratio of the numerator to the denominator: and therefore if the logarithms of the numerators and denominators will have the same differences, the terms from the definition of logarithms they will be proportionals, and the fractions themselves are equal.

*Literally speaking this is minus: as in fact it should be subtracting the defective [i.e. negative ] logarithm from the abundant [positive logarithm]. (The same) as by addition, expressing the usual commonly held view.

We will be able also to find the root of any given fraction if we look for the root of the numerator and the denominator. As the cube root of ⁸/₂₇ is ²/₃ .
As the homogeneous powers of the terms given, so are the terms of the roots being sought. As the fraction ⁸/₂₇ , the cube root is ²/₃ of which the numerator is the cube root of the given numerator 8. And the denominator 3 in like manner is the cube root of 27.
If the square root of ⁷²⁹/₄₀₉₆ is required:

For if of the fraction any other number we seek in the same series, with the positions of the roots, being multiplied by the logarithm of the root for the rational distance from unity: it will have made the logarithm of the homogeneous (i.e from the same root) number being sought. When if the given fraction is ³¹²⁵/₁₆₈₀₇ , I wish to know from four continued proportions between the given fraction and unity which will be the third from unity (?). But when there are four means, there will be five intervals, and the fifth part of the logarithm of the given fraction. will be the logarithm of the root, which triplicated will be the logarithm of the number from unity of the third (proportional). The whole operation you have lying here before your eyes.

If the seventh from unity is being sought, the logarithm of the root is multiplied by 7, the product D - 1,02289,62497,4768 will give ^10000000/_10541304 or ⁷⁸¹²⁵/₈₂₃₅₄₃ .
And this method of ours will give any number in the same series with the given number. For if we wish to know another number in the same series by continuing above one, the same logarithm of the root being multiplied, by the number of the interval between that sought and unity: the product will be the logarithm of the number sought. In fact the same will be the logarithm equidistant from unity and the other. But this lies between equal (logarithms), because the logarithm of the number above unity is abundant (i.e. positive); (that) below unity truly deficient (i.e. negative).
And as the logarithms on either side are equidistant, being written with the same places: thus with the same terms being transposed on either side they express the same absolute number. As with these numbers of which we have been most recently inquiring: ⁵/₇ and ⁷/₅ are close on either side to unity. The cubes or the fourth powers are [Briggs writes them in this order] ¹⁰⁰⁰/₂₇₄₄ and ²⁷⁴⁴/₁₀₀₀ , or ¹²⁵/₃₄₃ and ³⁴³/₁₂₅ . And unity is always the mean proportional between those equi-distant on either side for the same series. And in this way with numbers from unity in continued proportion, we shall be able from fractions to increase to whole numbers [i.e. improper fractions], and conversely with whole numbers to decrease to fractions.