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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
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
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
tan 2 a--
This rotation corresponds to the substitution
* ' 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