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THE 

AMERICAN 
MATHEMATICAL MONTHLY. 

Entered at the Post-office at Springfield, Missouri, as second-class matter, 

VOL. XV. JANUARY, 1908. NO. 1. 

NOTES ON THE HISTORY OF THE SLIDE RULE.* 



Br FLORIAN CAJORI, Colorado College. Colorado Springs. 



Few instruments designed for minimizing mental labor in computation 
offer a more attractive field for historical study than the slide rule. Its 
development has reached in many directions and has attracted a great vari- 
ety of intellect. Not only have writers on arithmetic been drawn to it, but 
also carpenters, excise officers, practical engineers, chemists, physicists, and 
mathematicians, including even the great Sir Isaac Newton. 

And yet, the history of this instrument has been neglected to such an 
extent that gross inaccuracies occur in standard publications. 

The first point I desire to make relates to the invention of the straight- 
edge slide rule. One of our American manufacturers of slide rules has an 
instrument on the market, called the "Gunter slide rule" and claims that it 
"is the original form of the slide rule." As a matter of fact Gunter never 
invented a slide rule. What Gunter did do was to publish in 1620, six years 
after Napier's publication of his logarithms, a work containing a description 
of Gunter's "line of numbers," which, when mounted upon a scale was 
called "Gunter's scale." On it distances were taken proportional to the 
logarithms of numbers; it was logarithms laid off upon straight lines. But 
Gunter's scale contained no sliding parts and, therefore, was not a slide rule. 

Charles Hutton, in his Mathematical Dictionary (Art. "Gunter's 
Line"), and also in his Mathematical Tables, ascribes the invention of the 
slide rule to Edmund Wingate, 1627, but he nowhere substantiates his 
statement by reference to any of Wingate's works. De Morgan in his 
article "Slide Rule" in the Penny Cyclopaedia (1842), and in later publica- 
tions, ascribes the invention to William Oughtred, a famous writer of 
mathematical text-books, 1632, and denies that Wingate ever wrote on the 
slide rule. It will soon appear that De Morgan was ill-informed on this 
subject, for he had not seen all of Wingate's works, although his criticism 
of a passage in Ward's Lives of the Professors of Gresham College (1740) is 
well taken. Ward claims that Wingate introduced the slide rule into France 

'Read before the Southwest Section of the American Mathematical Society, in St, Louis, November 30, 1907. 



in 1624. What he at that time really did introduce was Gunter's scale, as 
appears from the examination of his book, published in Paris in 1624. To 
prove or disprove the claim made for Wingate requires the examination of 
his numerous writings. To the present writer Wingate's publications are 
not accessible. An inquiry directed to the Keeper of the Printed Books at 
the British Museum in London brought the reply that in the work entitled, 
the Construction and Use of the Line of Proportion, London, 1628, the slide 
rule is explained. Prefixed to the book is a diagram of the "line of propor- 
tion," now called slide rule. Wingate says in his preface, "I have invented 
this tabular scale or line of proportion." Further on he says "the line of 
proportion is a double scale, broken off in tenne Fractions, upon which 
Logarithms of numbers are found out. " This book was probably reproduced 
two years later in Wingate's work Arithmetic made easy, or natural and 
artificial arithmetic, London, 1630, a text quoted by Favaro* in his history 
of the slide rule. A second edition of Wingates publication of 1630 appeared 
in 1652, wherein improvements in the divisions of the slide rule are 
described. From these facts it appears that De Morgan was in error, and 
that the claim made for Edmund Wingate as the inventor of the straight- 
edge slide rule is well founded, for he published four years earlier than did 
William Oughtred. It should be added, however, that Oughtred describes 
also a circular slide rule and that he has a clear title as the inventor of the 
circular type. 

My second point relates to the invention of the "runner." In 1850 a 
French artillery-officer and mathematician, A. Mannheim, designed a slide 
rule with a "runner," now generally known as the "Mannheim rule."' 
German writersf have called attention to the fact that Mannheim was not 
the first inventor of the runner, that a description of it occurs in a French 
work of 1837, thirteen years earlier. My own reading reveals that the run- 
ner was invented much earlier in England and afterwards completely for- 
gotten by the English. The first traces go back to Sir Isaac Newton, but in 
1842 even De Morgan who writes at length on the slide rule and its history, 
makes no reference whatever to the runner. It is not generally known that 
Sir Isaac Newton referred to the slide rule. In Newton's works is given an 
extract from a letter of Oldenburg to Leibniz, dated June 24, 1675, which 
we shall consider more fully later. The "runner" is not mentioned in this 
extract, but Newton's slide rule could not be used without the employment 
of some device like that of the ' 'runner. " Sixty-eight years later, Newton's 
scheme slightly modified is explained more fully in Stone's Mathematical 
Dictionary, 2nd Ed., 1743. I am not aware that Newton's and Stone's slide 
rules were ever actually constructed and used in practice. But thirty-five 
years after Stone's publication a book was published in London, containing 

* Veneto Institute Atti (6) B, 1878-79, p. 495, abbreviated in Favaro's Learns de statique graphique, 2 erne par- 
tie, calcid trraphique, Paris, 1885, translated into the French by P. Terrier. 
iZeitschriftf. Math, and Phys., Vol. 48, 1908, p. 134. 



an instrument by John Robertson, which employed the runner and which 
was constructed in Cornhill by Messrs. Nairne and Blunt, and put upon the 
market There are no indications that Robertson's rule ever became popu- 
lar. Later the use of the runner was advocated by William Nicholson in an 
article printed in the Philosophical transactions of 1787. But in the first 
half of the nineteenth century I have not been able to find a single reference 
to the "runner" in England. It was completely forgotten. 

Returning to Newton, I shall take up my third point, the early use of 
the slide rule in the solution of numerical equations. Oldenburg's letter to 
Leibniz, previously referred to, reads in translation from the Latin as fol- 
lows: "Mr. Newton, with the help of logarithms graduated upon scales by 
placing them parallel at equal distances or with the help of concentric circles 
graduated in the same way, finds the roots of equations. In the arrange- 
ment of these rules all the respective coefficients lie in the same straight 
line. From a point of which line, as far removed from the first rule as the 
graduated scales are from one another, in turn, a straight line is drawn over 
them, so as to agree with the conditions conforming with the nature of the 
equation; in one of these rules is given the pure power of the required root." 

If my interpretation of this passage is correct, it means in the case of 
the cubic x 3 +ax* +bx=c that the rules A, B, D must be placed parallel and 
equidistant On rule A find the number equal to the numerical value of the 
coefficient a; on rule B find the number equal to the numerical value of b, 
and on rule D find unity. Then arrange 
these three numbers on the rules in a 
straight line BD. Select the point E on 
this line, so that BE^BA. Through E \ 
pass a line ED' and turn it about E until 
the numbers at B', A', and D', with their 
proper algebraic signs attached, are seen j 
to be together equal to the absolute term c. 
Then the number on the scale D' is equal to | x 3 | , and x can be found. 

Remembering that the length of B"B is log | b | , and assuming BB'= 
log | x | , it follows that B'B' is equal to log | bx | . Then AA'=2\og | x | , or 
log | x 2 I and A"A'=\og | ax* | , and DD'=\og | a s | . The value of x can 
be found by moving the scale B up until B" reaches the point B. The num- 
ber on the scale at B' will then give the numerical value of the root. A 
device, as represented by the line ED', fulfills some of the functions of what 
is now called the "runner." 

In Stone's Dictionary (1743) Newton's scheme is modified somewhat 
Stone assumes that the equation is so transformed that all its coefficients, 
except the absolute term, are positive, The rules are contiguous and are not 
all graduated alike, but have, respectively, a single, double, triple, quad- 
ruple, etc., radius. This device calls for a runner of the type now in use, 
carrying a thread that is at right angles to the rules. Otherwise the gen- 




eral plan for the numerical solution of equations is the same as with 
Newton. 

As a fourth point in the history of the slide rule, I desire to point out 
that, while so generally known to writers on the slide rule, the English 
astronomer William Pearson was the first one to suggest, in 1797, the inver- 
sion of the slider for certain operations with the slide rule, the inversion of 
fixed lines on the slide rule had been introduced more than one hundred 
years earlier in Everard's slide rule, used in gauging. 

Finally, I desire to say a word as to the introduction of the slide rule 
into the United States. Brief directions for the use of the slide rule 
appeared in a few arithmetics imported, or reprinted in this country, in the 
latter part of the eighteenth century. Thus, the Arithmetic of. George 
Fisher, which is a pseudonym for Mrs. Slack, probably the first woman who 
is the author of a popular arithmetic, contained rules for the use of the slide 
rule. Her books were read in the United States. In Nicolas Pike's arith- 
metic, an American text of 1788, such rules were given. An edition of the 
English book, Dilworth's Schoolmaster's Assistant, was brought out in 
Philadelphia in 1805 by Robert Patterson, professor of mathematics in the 
University of Pennsylvania. It devotes half a dozen pages to the use of the 
slide rule in gauging. Another English work, Hawney's Complete Measurer 
(1st English Edition, 1717), was printed in Baltimore in 1813. It 'describes 
the English carpenter's rule, also an English rule for gauging. Of Ameri- 
can works, Bowditch's Navigator, 1802, gives one page to the explanation 
of the slide rule, but when working examples, Gunter's line alone is used. 
From these data it is difficult to draw reliable conclusions as to the extent to 
which the slide rule was then actually used in the United States. We sur- 
mise that it was practically unknown. The Swiss geodesist, F. R. Hassler, 
who came to this country and became the first superintendent of the United 
States Coast and Geodetic Survey, is known to have used a slide rule. The 
present writer had the good fortune, of inspecting Hassler's slide rule. But 
before 1880 or 1885 it is very difficult to find references to the slide rule in 
American engineering literature. I have seen a reference to the slide rule 
in a book issued in the first half of the last century by a professor of the 
Rensselaer Polytechnic Institute. From this institute was graduated in 1863 
Mr. Edwin Thacher, a bridge engineer, who in 1881 patented his well-known 
cylindrical slide rule. Interest in slide rules was awakened about this time. 
It was in 1881 that Robert Riddell published in Philadelphia his booklet on The 
Slide Rule Simplified. In the preface he points out that, though nearly 
unknown in this country, the instrument was invented before the time when 
William Penn founded Philadelphia. But the slide rule never became really 
popular in the United States until the introduction of the Mannheim type. 
Keuffel and Esser imported Mannheim rules in 1888 and began the manu- 
facture of them in Jersey City in 1895. An inquiry* instituted by C. A. 

*Engineering News, Vol. 45, 1901, p. 405. 



Holden in 1901 showed that in about half of the engineering schools of the 
United States, attention is given to the use of the slide rule. 



A BIQUADRATIC EQUATION CONNECTED WITH THE REDUCTION 

OF A QUADRATIC LOCUS. 



By DR. ARTHUR C. LUNN, The University of Chicago. 



If the equation of a conic section be written in the form 

Ax* +By 2 +2Cxy+2Dx+2Ey+F=0, 

then it is known that a rotation of the coordinate axes through an angle a 
will bring them into parallelism with the axes of symmetry of the curve, 
provided this angle is determined by 

2C 
tan 2 a-- 



A-K 
This rotation corresponds to the substitution 

,h\ x=x'cosa—y'sma, 

* ' y=y'cosa + a'sina, 

with a so chosen as to eliminate the term in x'y. But the sine and cosine 
may be expressed in terms of the tangent of the half -angle, thus: 

(2) ^tan-g-, cosa=jq^j, sma^jq^, 

and the use of these in (1) gives the substitution expressed rationally in 
terms of the parameter t. Without reference to its trigonometric source, the 
substitution in that form is seen to be orthogonal or rotational for all values 
of t, since the equation of constancy of distances: 

(x' i -x' s y + (y' 1 -y' s y=(x 1 -x i y + (y 1 -y i )\ 

is directly verifiable as an identity in t. 

The use of this parameter makes it possible to effect the reduction of 
the conic by purely algebraic processes, independently of the trigonometric 
formulae. For the term in x'y' will have as coefficient