Hutton (14 August 1737 – 27 January 1823)
Hutton was born in Newcastle, the youngest son of an
overviewer (supervisor) of a coal mine. When he was seven, Hutton was involved
in a street-brawl and severely dislocated his left elbow. He hid this injury
from his parents and by the time they learned of it, it was too late to treat it
properly, so the injury became permanent. Since Hutton was unable to join his
older brothers in the mine, he was sent to school to learn to read. After
several years the teacher left and Hutton replaced him, thus beginning a habit
of teaching by day and learning by night.
One pupil that Hutton attracted was Robert Shafto. He
made his private library available to Hutton and then encouraged him to publish.
Hutton’s first work, The Schoolmaster’s Guide, or a Complete System of
Practical Arithmetic, appeared in 1764 and became the standard school text for
half a century. During the Christmas holiday of 1666, Hutton advertised that
“Any schoolmaster, in town or country, who are desirous of improvement in any
branch of the mathematics, by applying to Mr Hutton, may be instructed” [Howson,
p. 63]. This in-service training was repeated the next year. That there was
ample audience is attested to by the 59 schoolmasters from the Newcastle area
who were subscribers to his next book, A Treatise on Mensuration (1767). Besides
its mathematical interest this work is noted for the woodcuts by the young
Thomas Bewick, who became one of the great masters of the woodcut. Alas, this
just makes the book more expensive for the historian of mathematics to acquire.
In 1760, Hutton opened his own school in Newcastle.
This became a success and he became known as an excellent teacher. His patron,
Shafto, suggested that he should move to London and apply for a vacancy at the
Royal Military Academy in Woolich. The position was to be filled by competitive
examination. Bishop Horsley, the editor of Newton’s works, and Nevel Maskelyn,
the Astronomer Royal, examined the eleven candidates. Half were judged
satisfactory for the post, but Hutton stood out, so he obtained this
professorship in 1773. He remained
at Woolich for 34 years.
Howson so nicely tells one event in Hutton’s career
that I shall quote the passage in its entirety:
In 1786 Hutton began to suffer from pulmonary disorders. The RMA was situated near the river and dampness began to affect is chest; his predecessor Simpson had in fact died from a chest complaint. Hutton decided then to move, and bought land on the hill south of the river overlooking Woolich. There he built himself a house and also others for letting. No sooner had he done this than it was decided to move the Academy from the damp riverside to the hilltop. A magnificent new building was erected, but, in the eyes of George III, its attractiveness was spoiled by the presence of Hutton’s houses. These were therefor sold to the crown who promptly demolished them, leaving Hutton with a hefty profit from his speculation, sufficient to guarantee his financial future. Thus a physical disability turned him to mathematics and ill-health made him rich. [Howson, pp. 66-67]
Hutton’s most important work was his Mathematical
and Philosophical Dictionary. This appeared in two volumes in 1795. The USMA
library has it, but the first volume is not the first printing. There is a
letter from Swift, saying that he read it before it was sent on to USMA. Hutton
worked on this for 10 or 12 years. It is an excellent survey of mathematics,
includes biographies of many mathematicians, and is a pioneer contribution to
the history of mathematics. [Howson, p. 67]. Margaret Baron writes that the Dictionary
I should try to find some reviews of this work. Make
an overhead of the title page.
Hutton is also famous as editor of The Lady’s
Diary, a journal that appeared from 1704 to 1841. See Perl 1979 for details.
His A Course in Mathematics was lauded before
it appeared. In its `Notices of works in hand’ the Monthly Magazine
(August 1798) stated:
From Dr H’s talents and long experience in his profession, there is every reason to expect that this will not only be a most useful and valuable work, but will completely supersede every other of the same description. [Howson, p. 67]
It did prove to be popular, appearing in numerous
editions over fifty years. There were several editions that were published in
North America and there was even an Arabic edition.
TO DO: Prepare a
bibliography of these editions. Pay especial attention to the US editions. Did
USMA play any role in their appearance? Where else did they appear? Where else
were they used? Do I have info about USMA buying copies of these? Did every
cadet get a copy?
It is not surprising that this text was used in the
US and at USMA, for the British influence on American education was extremely
strong at this time.
I have a copy of this work and need to include a
synopsis of what it includes. Also need to try to find out what portions of it
were actually taught at West Point. Are the copies in the library marked up?
Although earlier editions are available in the
library, I am working with the third American edition of 1822, since I happen to
own a copy of that edition and so have ready access. This edition is “revised,
corrected and improved” and contains “An Elementary essay on Descriptive
Geometry” by Robert Adrain. Naturally, I need to check later that the
arguments given below are not dependent on the edition that I am using. I must
point out, undoubtedly to the duress of my dear friend Danny Otero, that my copy
bears the “Sigilium St. Xavier” on the title page. Perhaps that is visible
in my overhead.
The first of the two volumes
consists of 533 pages! The main
topics are arithmetic, logarithms, algebra, and geometry. Here is a sample from
Multiplication is a compendious method of Addition, teaching how to find the amount of any given number when repeated a certain number of times ; as, 4 times 6, which is 24.
number to be multiplied, or repeated, is called the Multiplicand.
--- The number you multiply by, or the number of repetitions, is the Multiplier.
--- And the number found, being the total amount, is called the Product.
--- Also, both the multiplier and multiplicand are, in general named the Terms
Before proceeding to any operations in this rule, it is necessary to
learn off very perfectly the following Table, of all the products of the first
12 numbers, commonly called the Multiplication Table, or sometimes
Pythagoras’s Table, from its inventor.
I find --- and many before me agree --- the
exposition to be quite clear. On the other hand, his capitalization and
punctuation are of another era. His history is deplorable, even if it might (I
don’t know) represent his age.
The section on arithmetic, which consists of 154
pages, progresses to a discussion of square and cube roots, arithmetical and
geometrical progressions, compound interest, double position and permutations
and combinations. Since your study of arithmetic is somewhat in the past,
perhaps Hutton’s definition that “Double Position is the method of resolving
certain questions by means of two suppositions of false numbers.” (p. 137)
will be helpful. In a footnote he
provides a demonstration of the rule using algebra. So this is no lightweight
The section on algebra begins on p. 171 with a snarl
of definitions and notation, but a list of 26 simple examples of the use of the
notation immediately follows (e.g., if a = 6, b = 5, and c = 4, then 9ab –
10b^2 + c = 24 (#11, p. 174)). By page 213 he is discussing infinite series in
such problems as “To change 2ab/(a + b) into an infinite series.” The whole
book is, as was common in the period, rule based:
Rule IV. When the unknown quantity is included in any root or surd ; transpose the rest of the terms, if there be any, by Rule 1 ; then raise each side to such a power as is denoted by the index of the surd ; viz. Square each side when it is the square root ; cube each side when it is the cube root ; &c. which clears the radical. [p. 233]
Rules such as this are foreign to our texts today,
but the statement is not far removed from what we would say in class were we
teaching this material today. Part of what this reveals is that Hutton’s Course
was written for self study.
Hutton’s treatment of quadratic equations (pp.
249-257) has shades of Al-Khwarizmi (ca. A.D. 875) when he says that all
quadratic equations, when reduced, fall into one of the following forms:
x^2 + ax = b
x^2 – ax = b
x^2 – ax = -b
On a more positive side, few modern texts reveal how
“To resolve Cubic Equations by Carden’s Rule,” even if Hutton has trouble
spelling Cardano’s name.
The geometry section (pp. 275-368) begins in an
Euclidean fashion with
1. A POINT is that which has position, but no magnitude, nor dimensions ; neither length, breadth, nor thickness.
But then continues with another 76 definitions before
giving 11 axioms, none of which is the parallel postulate. The sequence of
Theorems which follow are very Euclidean, even if the usual sequence is not
preserved; for example, the Pythagorean Theorem, sans this common denomination,
is Theorem XXXIV (p. 300).
After a discussion of surveying, and “artificer’s
work,” i.e., carpenters, bricklayers, plasters, painters, and plumbers, there
is a section on the conic sections (pp. 469-534) that would severely tax any
These brief comments give some idea of what is
treated in the first volume. I will save a discussion of the second, which
contains a section on fluxions, until I learn which portions of this work might
actually have been taught to cadets at West Point. Unfortunately, I have not yet
located any documents that detail this. There are numerous sources that mention
“Hutton’s Course in Mathematics,” but none which detail the contents that
have been taught. There is a manuscript in the USMA library by one Cadet Wadell
Baron, Margaret, "Hutton, Charles," Dictionary of Scientific Biography.
Howson, A[lbert] Geoffrey, A History of Mathematics Education in England, Cambridge: Cambridge University Press, 1982. [Rickey owns a copy]
This contains a nice chapter on Charles Hutton (pp. 59-74). There is scant info about his boyhood, info on the schools he ran, and quite a bit of info on the books he wrote. He was not an original writer but a great expositor. There is also info on his career at The Royal Military Academy, Woolich, and his editorship of The Ladies Diary. His A Course in Mathematics, which reflects the curriculum at the Royal Military Academy in Woolich, England, was used as a text at USMA from 1802 to 1823.
Perl, Teri, "The Ladies' Diary or Woman's Almanack, 1704--1841," Historia Mathematica, 6 (1979), 36-53.
This annual was one of the most widely read popular journals of mathematics. Baron's The Mathematical Correspondent is modeled on it. Reviewed in Mathematical Reviews by : 82i:01024. See MathSciNet for the full review by Richard L. Francis.