Henry Briggs

Source: Henry Briggs, Arithmetica Logarithmica, Adrian Vlacq (ed.), Gouda, 1628. Translation by Daniel E. Otero.
 

    Logarithms are numbers which preserve equal differences with respect to proportionals.1
    Given any numbers whatsoever, other numbers could be identified different from these which, not inappropriately, will also satisfy the general definition of Logarithms and which could be of some welcome use.  If there be numbers 1, 2, 4, 8, 16, 32, 64, 128 in continued proportion, there can be associated to them as Logarithms numbers, denoted A or B or C or D or, as you will see here, others as well, subject to a single criterion: the differences of Logarithms by the same increment or decrement are equal to the number of times that the numbers to which they are associated are proportional.  Therefore it is not incorrectly said that Logarithms are equidifferent auxiliary numbers of proportional numbers.  For this reason they are named Logarithms by that most distinguished Inventor because they exhibit for us numbers maintaining the same ratio amongst themselves. 2
 
 

Let the Logarithm of unity be  0.  Although it is possible to construct in this way many different kinds of Logarithms for numbers, there is nonetheless exactly one of these ways that is most convenient, in which zero is put as the Logarithm of unity.  For this choice above all others offers everyone the most flexibility as well as the greatest of ease.  In support of this proposal, three most important axioms follow below.
    First.  Logarithms, or rather, those numbers which are called Indices, will exist for all numbers, and they are typically associated in all Arithmetics to numbers in continued proportions beginning with the unit and are set forth for a distance from the unit.  Alternatively, they are proportional to the usual Indices.
 
 

    A are numbers continually proportional from unity.  B are common Indices.  C are Logarithms proportional to these Indices.  In fact, when Logarithms, like Indices, for numbers continually proportional from unity (to which only the Indices are usually associated), increasing equally for a certain distance from an initial value, are proportional to their corresponding intervals, then they are also necessarily proportional to each other, as we see in the above numbers.
    Second.  The Logarithm of a product is equal to the [sum of the] Logarithms of the factors. 3   From the law of multiplication, there can always be set up a ratio between unity and multiplicand which equals that of multiplicator and product.4   This, together with the defintion of Logarithmsóthat proportionals have equidifferent Logarithms, makes clear from the second Lemma that the [sum of the] first and fourth Logarithms (that is, unity and that of the product) is equal to the [sum of the] Logarithms of the second and third (that is, the multiplicand and multiplicator).  And since the Logarithm of unity is 0, it is manifest that the Logarithm of the product alone is equal to the sum of the Logarithms of the factors 5, as we see below:
 
 

    If there are many factors, then the sum of all their Logarithms equals the Logarithm of their product.  For if the factors are 2, 8, 64, the number made continually from these is 1024, whose Logarithm, 3,01030, equals the sum of the Logarithms of the three of them, from whose multiplication this product was produced, as we see here:
 
 

    For since the Logarithm of the number E, 16, the product of A multiplied into B, equals the sum of the Logarithms of A and B, and for the same reason, the Logarithm of the number D equals the sum of the Logarithms of E and C, it is necessary that the Logarithms of the number D equal the sum of the Logarithms of the numbers A, B, and C, which is what we wished to prove.
    Third.  The Logarithm of the dividend equals the sum of the Logarithms of the divisor and quotient. 6   Indeed, this necessarily follows from the preceding, as the product of the divisor by the quotient is equal to the dividend.  Therefore, as unity is the divisor, so the quotient is to the dividend; so if the dividend were 128, the divisor were 4, the quotient would be 32.
 
 

    If after completing a first division, the Quotient becomes a new dividend, the Logarithm of the first dividend will be equal to the sum of the Logarithms of both divisors and the next quotient, as the second divisor and quotient can be substituted in place of the first quotient.  So if the dividend were 105, and divisor 7, the quotient would be 15, which if we divide by 5, the quotient is 3.  I claim that the Logarithm of the number 105 equals the sum of the Logarithms of the numbers 7, 5, 3, as we see here:
 
 

    Further, these three properties necessarily follow if we establish that the Logarithm of unity is zero.
 
 

    Having established the Logarithm of unity, we next find that of another number which is frequently used and is most indispensible, to which we assign the Logarithm to ensure such a measure of convenience that it will be both most easy to describe and easy to remember.  Indeed, of all numbers, none is seen to be more fundamental to this endeavor than Ten, whose Logarithm is 1,00000,00000,0000.7
    I assign therefore to the special numbers Unity and Ten the Logarithms 0 and 1,00000,00000,00000.  We have first chosen these particular four numbers, not for some necessary reason, but arbitrarily, with no end other than facility of use, having considered the certitude of the operations of Arithmetic (but which could have been obtained by many other and more diverse means).  The remaining Logarithms (when  determined, they will be turn out not to be arbitrary integers at all) are thus all dependent on these first, wherefore, by total agreement everywhere, from beginning to end, they obey these laws, and can be computed as required by use.
    Of those Logarithms which we seek, some are rational which can be found and displayed exactly, and some are irrational which although they cannot in any way be presented completely, they can be presented approximately and really so that barely one unit in a large number will fall short or exceed it.
    Rational Logarithms.  Rational Logarithms are assigned not just to the two values Unity and Ten, on which the structure of the all remaining values depends fundamentally, but also to every other value that can be thrown together in any manner whatsoever with these in some continually proportional series, as you can see here:
 
 

    A are numbers continually proportional from Unity.  C are the absolute values 8  of these in numbers, which are also in continued proportion. B are the rational and true Logarithms of these.
    All remaining numbers that do not lie in any series in continued proportion (in which the given two numbers Unity and Ten also lie) have irrational Logarithms which can be accurately expressed not just with integers but with parts, which are called fractions.  Although we cannot obtain them exactly, we can nonetheless determine them so nearly correctly that if we were to compare the benefits of both rational and irrational ones, nothing would distinguish them.  Ptolemyís Table of Chords suffered under this same problem, as did all subsequent Tables of Sines, Tangents and Secants of Astronomers down through time, without any serious inconvenience.
 
 

[...]
   Let the Cube 979 be given: we desire the side of the Cube.9  The Logarithm of the given is 2,99078,26918, from which we extract the third part: 0,99692,75936.  In this example, as in the previous, the Characteristic (which we took in Chap. 4 to be the leftmost digit) is carefully considered.  To wit, since a third part of the first digit cannot be taken,10 I place 0 in the first place in the quotient for its Characteristic, which shows that the desired side is less than ten.
    Now since the side cannot be found amongst the integers11, we consider the desired side to be multiplied by 10000, whose Logarithm is 4,00000,00000, and I add it to the third part, the total being 4,99692,75936.  This corresponds to 99295 and a bit, from the last Chiliad.12  Whence for the sake of accuracy, by differences and proportional part, can be found 9929504202; that is, 9^29504202/_100000000, which is Nine with some parts adjoined.
    By this method then the side of any proposed power can be obtained, if not exactly then nearly so.
[...]
 
 

    Equipped with this method, suppose we want to know how much should be collected on the loan after seven years, five months and nine days: each of the differences should be multipled by its number and the sum of the products added to the Logarithm of the principal; the sum will be the Logarithm of the combined principal and interest, determined for the end of the period.13
 
 

If this sum is subtracted from the Logarithm of the principal, the remainder will be the Logarithm of the just price for which the loan should be presently redeemed in cash where we have counted how many years, months, and days occur before the day of disencumbrance.  Consider the example.
    I claim that 123 pounds at the end of 7 years, 5 months and 9 days is worth 189⁷⁶⁴²/₁₀₀₀₀, that is, 189-15-3⁴¹/₁₀₀.  [Trans.: This is expressed in English currency of the day: pounds-shillings-pence; there were 20shillings to the pound and 12 pence to the shilling.]  And if on the day of disencumbrance 123 pounds are released, the just price of redemption for the number of years, months and days until that time is represented in cash by 79⁷²⁵³/₁₀₀₀₀, or 79-14-6⁶/₁₀₀.  Here the ratio of the interest is maintained: the principal 79-14-6⁶/₁₀₀ after 14 years, 10 months and 18 days is worth 189-15-3⁴¹/₁₀₀.
    This same procedure holds given any other ratio of principal to interest for any given period of time, either before or after the day of disencumbrance.
 
 

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