A History of Elementary Mathematics; With Hints on Methods of Teaching

📗 This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1917 edition. Excerpt: ...moments are, therefore, represented by the first of the two following series (approximately), 0, 1, 2, 3,-., v. The second series represents at the end of corresponding intervals of time very nearly the distances of D' from A'. According to Napier, the numbers in the lower series are approximately the logarithms of the corresponding numbers in the upper series. Now observe that the lower series is an arithmetical progression and the upper a geometrical progression. It is here that Napier's discovery comes in touch with the work of previous investigators, like Archimedes and Stifel; it is here that the continuity between the old and the new exists. The relation between numbers and their logarithms, which is indicated by the above series, is found, of course, in the logarithms now in general use. The numbers in the geometric series, 1, 10, 100, 1000, have for their common logarithms (to the base 10), the numbers in the arithmetic series, 0, 1, 2, 3. But observe one very remarkable peculiarity of Napier's logarithms: they increase as the numbers themselves decrease and numbers exceeding v have negative logarithms. More over, zero is the logarithm, not of unity (as in modern logarithms), but of v, which was taken by Napier equal to 107. Napier calculated the logarithms, not of successive integral numbers, from 1 upwards, but of sines. His aim was to simplify trigonometric computations. The line AE was the sine of 90 (i.e. of the radius) and was taken equal to 107 units. BE, GE, DE, were sines of arcs, and A'B', A'C', A'D' their respective logarithms. It is evident from what has been said that the logarithms of Napier are not the same as the natural logarithms to the base e = 2.718.... This difference must be emphasized, because it is not...