History of logarithms

The next improvement in mathematics, which we have to mention, is the introduction of logarithms, those numbers so important by diminishing the labour of tedious calculations, and which play so conspicuous a part in the transcendental analysis. For this admirable discovery we are indebted to John Napier, Baron of Merchiston, near Edinburgh. Napier seems to have turned the bent of his genius towards the discovery of methods to facilitate and abridge trigonometrical calculations; and various contrivances were proposed by him in succession, all remarkable for their ingenuity. The last and most memorable of all was his discovery of logarithms.

— History of logarithms success

There is a story told by Mr. Wood, but it does not appear entitled to any attention, that one Dr. Craig, a Scotchman, coming out of Denmark into his own country, called upon John Napier and told him of a new invention in Denmark, by Longomontanus, to save tedious multiplications and divisions in astronomical calculations. Napier being solicitous to know further of him of this matter, he could give no other account of it than that it was by proportional numbers; which hint Napier taking, desired him, at his return, to call upon him again. Craig, after an interval of some weeks, did so, and Napier then showed him a rude draught of what he called canon mirabilis logarithmorum. Had there been any truth in such a story, we may be sure that Longomontanus and the Danes would not have abstained from laying their claim to so admirable a discovery.

— History of logarithms truth

Napier has also been considered as having been anticipated in his invention by Stifels, and by Juste Byrge, two German mathematicians; but these allegations originating from jealousy, or from national partiality, are entitled to no attention whatever, and Napiers claims have for many years been allowed by the universal consent of all mankind.

— History of logarithms art

The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms. The invention of logarithms in the first quarter of the seventeenth century was admirably timed, for Kepler was then examining planetary orbits, and Galileo had just turned the telescope to the stars. During the Renaissance German mathematicians had constructed trigonometrical tables of great accuracy, but this greater precision enormously increased the work of the calculator. It is no exaggeration to say that the invention of logarithms by shortening the labours doubled the life of the astronomer.

— History of logarithms life

Logarithms were invented by John Napier It is one of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to Harriot, whose algebra appeared long after Napiers death. That logarithms flow naturally from the exponential symbol was not observed until much later. It was Euler who first considered logarithms as being indices of powers.

— History of logarithms death

What was Napiers line of thought? Napiers process is so unique and so different from all other modes of presenting the subject that there cannot be the shadow of a doubt that this invention is entirely his own; it is the result of unaided, isolated speculation. He first sought the logarithms only of sines

— History of logarithms sin

It is evident from this formula that Napiers logarithms are not the same as the natural logarithms. Napiers logarithms increase as the number itself decreases.

— History of logarithms self

Napiers genesis of logarithms from the conception of two flowing points reminds us of Newtons doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by Archimedes Stifel and others. Napier did not determine the base to his system of logarithms. The notion of a base in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napiers figures, owing to slight inaccuracies in the calculation of the tables.

— History of logarithms mind

The confusion marked in the writings of Halley and Saverien spread among French writers. Montuclu, the great mathematical historian of the eighteenth century, made the same mistake; Bossut helped to perpetuate the error.

— History of logarithms writing

De Morgan carefully explains the difference between Napiers and natural logarithms in the article Tables in the English Cyclopaedia but in De Morgans Budget of Paradoxes р. 70 Günther has found a passage which is inaccurate.

— History of logarithms art

It is a pleasure to find that Kästner presents the subject in a way free of error. In his Geschichte he refers to an article, which he had written, setting forth the exact relation between the two systems. Nevertheless the misconception became prevalent in Germany also.

— History of logarithms art

Speidells sole object was to simplify matters for persons unacquainted with the use of negative quantities. Speidell did not advance a new theory. He simply aimed to make all the logarithms in his table positive.

— History of logarithms positive

: From Its Institution to the End of the Eighteenth Century by Thomas Thomson, Book II. Of Mathematics

— History of logarithms book

The word logarithm means ratio number and was an afterthought with Napier. He first used the expression artificial number, but before he announced his discovery he adopted the name by which it is now known.

— History of logarithms art

Napier worked at least twenty years upon the theory. His idea was to simplify multiplications involving sines, and it was a later thought that included other operations, applying logarithms to numbers in general. He may have been led to his discovery by the relationfor, as Lord Moulton says, in no other way can we conceive that the man to whom so bold an idea occurred should we have so needlessly and so aimlessly restricted himself to sines in his work, instead of regarding it as applicable to numbers in general.

— History of logarithms work

In Napiers time sin φ was a line, not a ratio. The radius was called the sinus totus, and when this was equal to unity the length of the sine was simply stated as sinφ. If r was not unity, the length was r sinφ. With this statement we may consider Napiers definition of a logarithm:The logarithme therefore of any sine is a number very neerely expressing the line, which increased equally in the meane time, whiles the line of the whole sine decreased proportionally into that sine, both motions being equal-timed, and the beginning equally swift.From this it follows that the logarithm of the sinus totus is zero. Napier saw later that it was better to take log 1 = 0.

— History of logarithms time

Napiers logarithms are not those of the so-called Naperian, or hyperbolic, system, but are connected with this system by the relationThe relation between the sine and its logarithm in Napiers system isso that the sine increases as its logarithm decreases.

— History of logarithms sin

Henry Briggs was one of the first to appreciate the work of Napier. Upon reading the Descriptio he wroteNaper, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.

— History of logarithms god

Briggss Aritmetica Logarithmica the preface contains the following statement by the authorThat these logarithms differ from those which that illustrious man, the Baron of Merchiston published in his Canon Mirificus must not surprise you. For I myself, when expounding their doctrine publicly in London to my auditors in Gresham College, remarked that it would be much more convenient that 0 should be kept for the logarithm of the whole sine as in the Canon Mirificus And concerning that matter I wrote immediately to the author himself; and as soon as permitted I journeyed to Edinburgh, where, being most hospitably received by him, I lingered for a whole month. But as we talked over the change in logarithms he said that he had for some time been of the same opinion and had wished to accomplish it. He was of the opinion that 0 should be the logarithm of unity.

— History of logarithms time

The real value of the proposition made by Briggs at this time was that he considered the values of log 10n a, for all values of n. The relation between the two systems as they first stood were as follows: Napier, , where Briggs, Napier later suggestion,

— History of logarithms time

which are the fundamental laws of logarithms. Most writers refer to Stifel as the first to set forth these basal laws but he was by no means the first Probably the best statements concerning them which appeared in the 15th century were those of Chuquet in Le Triparty en la Science des Nombres Chuquet expressed very clearly the relationsand

— History of logarithms science

distinctly calling the upper numbers exponents, and saying that the exponents of the factors are added to produce the exponent of the product and subtracted to produce the exponent of the quotient. Moreover, he expressly lays down four laws, namely, that addition in arithmetic progression corresponds to multiplication in geometric progression, that subtraction corresponds to division, multiplication to the finding of powers, and division to the extracting of roots. Furthermore, Stifel not only set forth the laws for positive exponents but also saw the great importance of considering the negative exponents of the base which he selected, using the series

— History of logarithms power

and making the significant remark: I might write a whole book concerning the marvellous things relating to numbers but I must refrain and leave these things with eyes closed. What these mysteries were we can only conjecture.

— History of logarithms book

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