DIFFERENCE BETWEEN NAPIER'S AND NATURAL LOGARITHMS.*

The term "Napierian logarithms" has been used in three different senses: (1) as meaning Napier's logarithms, or the ones invented by him and published in 1614 in his Mirifici Logarithmorum Canonis Descriptio; (2) as a synonym for "natural logarithms;" (3) as conveying the first and second meanings combined, and, thereby, implying that the natural logarithms are the ones invented by Napier. Though this last use of the term is inadmissible, because the logarithms invented and published by Napier are really different from the natural logarithms, it has, nevertheless, been the most prevalent; especially has it been prevalent in this country.

An examination of the algebras which have been in use in our schools will at once convince us that this error has been very general. We may consult the algebras of Ray, Greenleaf, Ficklin, Schuyler, Loomis, Robinson, F. H. Smith, Hackley, Davies, Bowser, Stoddard and Henkle, Thomson and Quimby, and many others, and we find it stated either that Lord Napier selected for the base of his system e 2.718. . ., or that he assumed the modulus equal to unity. Either of these two statements is equivalent to saying that the logarithms invented by Napier are identical with the natural logarithms. Some authors make statements like the following one, taken from the revised edition of Wells's University Algebra (p. 363): "The system of logarithms, which has e for its base, is called the Napierian system, from Napier, the inventor of logarithms."

The objection to statements like this is that they almost invariably mislead the student. What inference is more natural than that Napierian logarithms were invented by Napier? Some explanation ought therefore to be made guarding against this error.

But I have seen only two American books doing this, namely, J. M. Peirce's Mathematical Tables, and Van Velzer and Slichter's Course in Algebra (of which a preliminary edition has just appeared). In these two books the truth is conveyed in plain words that Napier's logarithms differ from the natural. It is the object of this article to explain that difference.

It is important to note that, in Napier's time, our exponential notation in algebra had not yet come into use. To be sure, Stifel in Ger many and Stevin in Belgium had, previous to this, made some attempts at denoting powers by indices; but this notation was not immediately appreciated, nor was it generally known to mathematicians, not even to the celebrated Harriot, whose algebra appeared long after Napier's death. It is one of the greatest curiosities in the history of mathe

* This article has been published in the Mathematical Magazine, Vol. II, No. 1, and is here reproduced with some very slight changes.

matics that logarithms should have been constructed before exponents were used. We know how naturally logarithms flow from the exponential symbol, but to Napier this symbol was entirely unknown.

The interesting inquiry then arises, What was Napier's treatment of logarithms? It may be briefly stated as follows:

Let AB be a line of definite length, DE a line extending from D indefinitely. Imagine two points set in motion at the same time, and with the same initial velocity; the one point moving from D toward E with uniform velocity; the other from A to B with a velocity decreasing in such a way that when it arrives at any position, C, its velocity is proportional to the remaining distance, BC. While the latter point travels a distance, AC, suppose the former to move over the space DF. Napier called DF the logarithm of BC. He first applied this idea to the calculation of a table of logarithms for the natural sines in trigonometry. In the above figure, AB would represent the sine of 90° or the radius, which was taken by him equal to 10,000,000 or 107. BC would be the sine of an arc, and DF its logarithm.

"The logarithm, therefore, of any sine is a number very nearly expressing the line which increased equally in the meantime, while the line of the whole sine decreased proportionally into that sine, both mo tions being equal-timed, and the beginning equally swift."*

This treatment of the subject is certainly very unique. Let us now establish the relation between these Napierian logarithms and our natural logarithms. Let m=AB, x=DF, y=BC, then AC=m-y. The velocity of the point C is d(my)=ry, r being a constant. Integrat ing, we have

dt

-Nat log y=rt+c.

When t=0, then y=m, and c=-Nat log m. The velocity of the point C is rm, when t=0. Since the two points start with the same velcity, dx we have dt

=rm as the uniform velocity of the point F. Hence x= rmt. Substituting for t and c their values, and remembering that, by definition, x=Nap log y, we get

The constant m was taken equal to 107. Substituting we get

as the equation expressing the relation between Napierian and natural logarithms.

* Definition 6, p. 3, of Napier's Mirifici Logarithmorum Canonis Descriptio, etc., 1614.

That there is a difference between the two is evident at once. We easily observe the characteristic property of Napierian logarithms, that they decrease as the number itself increases. This property alone should have been a sufficient guard against declaring the two systems identi cal. The Napierian logarithm of 10 is equal to zero. The Napierian logarithms of numbers smaller than it are positive; those of numbers larger than it are negative, or, in the language of Napier, "less than nothing.". In further illustration we give the following:

Nap. log. 1

Nap. log. 2
Nap. log. 10

161 180 956.509; Nat. log. 1 = 0

154 249 484.703; Nat. log. 20.6 931 472
138 155 105.578; Nat. log. 10 = 2.3 025 851

The question may be asked what base Napier selected for his system. We answer that he did not calculate his logarithms to a base at all. He never thought nor ever had any idea whatever of a base in connection with logarithms. The notion of a base suggested itself to mathematicians later, after the algorithm of powers and exponents, both integral and fractional, had come to be better understood.

If we inquire what the base to the logarithms in Napier's tables would have been had he used one, then it will be found that it does not coincide with the natural base e, but is very nearly equal to its recip. rocal. In theory, that base is exactly equal to the reciprocal of e, as will be seen from the following relation,* which is merely another form of the one given above,

1

e

Nap log y
107

The base would not lead accurately to Napier's logarithmic figures, because the inventor's method of calculation was necessarily somewhat rude and inexact. The modulus of his logarithms is not equal to 1, but nearly equal to -1. If the base were exactly 1, then the modulus would be exactly -1; for the modulus of any system of logarithms is the logarithm, in that system, of the Napierian base e.

The first calculation of logarithms to the base of the natural system was made by John Speidell in his New Logarithms, published in London, in 1616, or five years after the first appearance of Napier's logarithmic tables.†

* To make the theory of exponents applicable to Napier's logarithms, it becomes necessary to divide the number y by 107, otherwise the base raised to the zero power would not be equal to unity. This division really amounts to making the length of the line AB equal to 1 instead of 107. If this be done, then Nap. log. y must also be divided by 10', so as to retain the inventor's conception that the two points on the lines AB and DE, respectively, move with equal initial velocities.

+ The error of calling the Napierian and natural logarithms one and the same system has been wide-spread. We may pardon the celebrated Montucla, the eldest prominent writer on the history of mathematics, for making this mistake (Montucla,

CIRCLE SQUARERS.

It would be strange if America had not produced her crop of "circlesquarers," just as other countries have done. Our history of them will be quite incomplete. We have not gone out of our way to seek the acquaintance of this singular race of "mathematicians," nor have we avoided them. A few individuals have come across our path, and we proceed to tell about them for the benefit and edification not so much of mathematicians as of psychologists. The mathematician contemplates the products of only sound intellects; the psychologist studies also the utterings of minds that are or seem to be diseased.

The history of the quadrature of the circle is not without its sober lessons to mathematicians. It extends back through centuries almost to the beginning of geometry as a science.

The student of the history of mathematics is impressed by the fact that this science, more than any other, has always been a progressive one. He does not find a period in authentic history during which mathematics was not cultivated quite successfully by some nation or other. The earliest contributions were made by the Babylonians and Egyptians, then came the Greeks, then the Hindoos, then the Arabs, and finally the Europeans. Like metaphysics, mathematics has encountered fundamental problems apparently of insurmountable difficulty. But it has generally had the good fortune to perceive that fortifications can be taken in other ways than by direct attack with open force; that, when repulsed from a direct assault, it is well to reconnoitre Histoire des Mathématiques, Tome II, Paris, 1758, p. 21), but there is hardly any excuse for a modern writer, such as Hoefer (Histoire des Mathématiques depuis leurs Origines jusqu'au Commencement du Dix-neuvième Siècle, Paris, 1874, p. 378), for stumbling over the same stone. The difference between the two systems was pointed out in Germany by Karsten in 1768, Kaestner in 1774, and Mollweide in 1808, but no attention was paid to their writings on this subject. A lucid proof of the nonidentity of the two systems was given by Wackerbarth ("Logarithmes Hyperboliques et Logarithmes Népériens," Les Mondes, Tome XXVI, p. 626). The French mathematician Biot wrote likewise on this subject (Journal des Savants, 1835, p. 259), as did also De Morgan in England (English Cyclopædia, Article "Tables"). Still more recently attention has been called to this matter by J. W. L. Glaisher (Encyclopædia Britannica, 9th ed., Article "Logarithms"), and by Siegmund Guenther ( UnterBuchungen zur Geschichte der mathematischen Wissenschaften, Leipzig, 1876, p. 271). The writings of these scientists do not seem to have received the attention they deserve, and the erroneous notion of the identity of Napierian and natural logarithms still continues to be almost universal.

Napier's original works on logarithms are very scarce. The Mirifici Logarithmorum Canonis Descriptio, etc., Edinburgi, 1614, can be found in the Congressional Library in Washington and in the Ridgway Library in Philadelphia. The latter library has also the English edition of the above work, translated by Edward Wright in 1616. "So rare are these original editions that, of the two greatest historians of logarithms, Delambre never saw the Latin edition and Montucla never heard of the English." (Mark Napier's Biography of Lord Napier, p. 379).

and occupy the surrounding country and discover the secret paths by which the apparently unconquerable position can be taken.*

From this we can draw the valuable lesson that it is not always best to "take the bull by the horns."

The value of this precept may be seen by giving an instance in which it has been violated. The history of the quadrature of the circle is in point. An untold amount of intellectual energy has been expended upon this problem, yet no conquest has been made by direct assault. The circle-squarers existed in crowds even before the time of Archimedes and in all succeeding ages in which geometry was cultivated, down even to our own. After the invention of the differential calculus abundant means were introduced to complete the quadrature, if such a thing were possible. Persons versed in mathematics became convinced that the problem could not be solved, and dropped it. But those who still continued to make attempts upon this "enchanted castle," as it was supposed to be, were completely ignorant of the history of the subject, and generally misunderstood the conditions of the problem. "Our problem," says De Morgan,† "is to square the circle with the old allowance of means: Euclid's postulates and nothing more. We can not remember an instance in which a question to be solved by a definite method was tried by the best heads and answered at last by that method, after thousands of complete failures."

But great advance has been made on this problem by approaching it from a different direction and by newly discovered paths. Lambert, an Alsacian mathematician, proved in 1761 that the ratio of the circumference of a circle to its diameter is incommensurable. Only nine years ago Lindemann, a German mathematician, demonstrated that this ratio is also transcendental, and that the quadrature of the circle by means of the ruler and compass only, or by means of any algebraic curve, is impossible. He has thus shown by actual proof that which keen-minded mathematicians had long suspected, namely, that the great army of circle-squarers have, for more than two thousand years, been assaulting a fortification which is as impossible to be torn down as the firmament of heaven is by the hand of man.

Now-a-days, a person claiming to have solved this problem is ranked by mathematicians in the same class with inventors of "perpetual motion," and discoverers of the "fountain of perpetual youth." A very peculiar characteristic of circle-squarers, or quadrators, as Montucla calls them, is that they cannot be convinced of their errors. The first American quadrator we shall mention is William David Clark Murdock, who, in a pamphlet of eight pages, bearing no date, gives a Demonstration of the Quadrature of the Circle.

The next man on our list is John A. Parker, whose work on The Quad

* H. Hankel, Entwickelung der Mathematik in den letzten Jahrhunderten, p. 16. + English Cyclopædia; article, "Quadrature of the Circle."

Mathematische Annalen, Band XX, p. 213.