PRINCIPAL WORKS
“CLAVIS MATHEMATICAE”
Passing to the consideration of Oughtred’s mathematical books, we begin
with the observation that he showed a marked disinclination to give his
writings to the press. His first paper on sun-dials was written at the
age of twenty-three, but we are not aware that more than one brief
mathematical manuscript was printed before his fifty-seventh year. In
every instance, publication in printed form seems to have been due to
pressure exerted by one or more of his patrons, pupils, or friends. Some
of his manuscripts were lent out to his pupils, who prepared copies for
their own use. In some instances they urged upon him the desirability of
publication and assisted in preparing copy for the printer. The earliest
and best-known book of Oughtred was his Clavis mathematicae, to which
repeated allusion has already been made. As he himself informs us, he was
employed by the Earl of Arundel about 1628 to instruct the Earl’s son,
Lord William Howard (afterward Viscount Stafford) in the mathematics. For
the use of this young man Oughtred composed a treatise on algebra which
was published in Latin in the year 1631 at the urgent request of a
kinsman of the young man, Charles Cavendish, a patron of learning.
The Clavis mathematicae,[22] in its first edition of 1631, was a booklet
of only 88 small pages. Yet it contained in very condensed form the
essentials of arithmetic and algebra as known at that time.
Aside from the addition of four tracts, the 1631 edition underwent some
changes in the editions of 1647 and 1648, which two are much alike. The
twenty chapters of 1631 are reduced to nineteen in 1647 and in all the
later editions. Numerous minute alterations from the 1631 edition occur
in all parts of the books of 1647 and 1648. The material of the last
three chapters of the 1631 edition is rearranged, with some slight
additions here and there. The 1648 edition has no preface. In the print
of 1652 there are only slight alterations from the 1648 edition; after
that the book underwent hardly any changes, except for the number of
tracts appended, and brief explanatory notes added at the close of the
chapters in the English editions of 1694 and 1702. The 1652 and 1667
editions were seen through the press by John Wallis; the 1698 impression
contains on the title-page the words: Ex Recognitione D. Johannis Wallis,
S.T.D. Geometriae Professoris Saviliani.
The cost of publishing may be a matter of some interest. When arranging
for the printing of the 1667 edition of the Clavis, Wallis wrote Collins:
“I told you in my last what price she [Mrs. Lichfield] expects for it, as
I have formerly understood from her, viz., £ 40 for the impression, which
is about 9½d. a book.”[23]
As compared with other contemporary works on algebra, Oughtred’s
distinguishes itself for the amount of symbolism used, particularly in
the treatment of geometric problems. Extraordinary emphasis was placed
upon what he called in the Clavis the “analytical art.”[24] By that term
he did not mean our modern analysis or analytical geometry, but the art
“in which by taking the thing sought as knowne, we finde out that we
seeke.”[25] He meant to express by it condensed processes of rigid,
logical deduction expressed by appropriate symbols, as contrasted with
mere description or elucidation by passages fraught with verbosity. In
the preface to the first edition (1631) he says:
In this little book I make known . . . . the rules relating to
fundamentals, collected together, just like a bundle, and adapted to
the explanation of as many problems as possible.
As stated in this preface, one of his reasons for publishing the book, is
. . . . that like Ariadne I might offer a thread to mathematical study
by which the mysteries of this science might be revealed, and direction
given to the best authors of antiquity, Euclid, Archimedes, the great
geometrician Apollonius of Perga, and others, so as to be easily and
thoroughly understood, their theorems being added, not only because to
many they are the height and depth of mathematical science (I ignore
the would-be mathematicians who occupy themselves only with the
so-called practice, which is in reality mere juggler’s tricks with
instruments, the surface so to speak, pursued with a disregard of the
great art, a contemptible picture), but also to show with what keenness
they have penetrated, with what mass of equations, comparisons,
reductions, conversions and disquisitions these heroes have ornamented,
increased and invented this most beautiful science.
The Clavis opens with an explanation of the Hindu-Arabic notation and of
decimal fractions. Noteworthy is the absence of the words “million,”
“billion,” etc. Though used on the Continent by certain mathematical
writers long before this, these words did not become current in English
mathematical books until the eighteenth century. The author was a great
admirer of decimal fractions, but failed to introduce the notation which
in later centuries came to be universally adopted. Oughtred wrote 0.56 in
this manner 0|56; the point he used to designate ratio. Thus 3:4 was
written by him 3·4. The decimal point (or comma) was first used by the
inventor of logarithms, John Napier, as early as 1616 and 1617. Although
Oughtred had mastered the theory of logarithms soon after their
publication in 1614 and was a great admirer of Napier, he preferred to
use the dot for the designation of ratio. This notation of ratio is used
in all his mathematical books, except in two instances. The two dots (:)
occur as symbols of ratio in some parts of Oughtred’s posthumous work,
Opuscula mathematica hactenus inedita, Oxford, 1677, but may have been
due to the editors and not to Oughtred himself. Then again the two dots
(:) are used to designate ratio on the last two pages of the tables of
the Latin edition of Oughtred’s Trigonometria of 1657. In all other parts
of that book the dot (·) is used. Probably someone who supervised the
printing of the tables introduced the (:) on the last two pages,
following the logarithmic tables, where methods of interpolation are
explained. The probability of this conjecture is the stronger, because in
the English edition of the Trigonometrie, brought out the same year
(1657) but after the Latin edition, the notation (:) at the end of the
book is replaced by the usual (·), except that in some copies of the
English edition the explanations at the end are omitted altogether.
Oughtred introduces an interesting, and at the same time new, feature of
an abbreviated multiplication and an abbreviated division of decimal
fractions. On this point he took a position far in advance of his time.
The part on abbreviated multiplication was rewritten in slightly enlarged
form and with some unimportant alterations in the later edition of the
Clavis. We give it as it occurs in the revision. Four cases are given. In
finding the product of 246|914 and 35|27, “if you would have the Product
without any Parts” (without any decimal part), “set the place of Unity of
the lesser under the place of Unity in the greater: as in the Example,”
writing the figures of the lesser number in inverse order. From the
example it will be seen that he begins by multiplying by 3, the
right-hand digit of the multiplier. In the first edition of the Clavis he
began with 7, the left digit. Observe also that he “carries” the nearest
tens in the product of each lower digit and the upper digit one place to
its right. For instance, he takes 7×4=28 and carries 3, then he finds
7×2+3=17 and writes down 17.
2 4 6|9 1 4
7 2|5 3
-------
7 4 0 7
1 2 3 5
4 9
1 7
-------
8 7 0 8
The second case supposes that “you would have the Product with some
places of parts” (decimals), say 4: “Set the place of Unity of the lesser
Number under the Fourth place of the Parts of the greater.” The
multiplication of 246|914 by 35|27 is now performed thus:
2 4 6|9 1 4
7 2|5 3
---------------
7 4 0 7 4 2 0 0
1 2 3 4 5 7 0 0
4 9 3 8 2 8
1 7 2 8 4 0
---------------
8 7 0 8|6 5 6 8
In the third and fourth cases are considered factors which appear as
integers, but are in reality decimals; for instance, the sine of 54° is
given in the tables as 80902 when in reality it is .80902.
Of interest as regards the use of the word “parabola” is the following:
“The Number found by Division is called the Quotient, or also Parabola,
because it arises out of the Application of a plain Number to a given
Longitude, that a congruous Latitude may be found.”[26] This is in
harmony with etymological dictionaries which speak of a parabola as the
application of a given area to a given straight line. The dividend or
product is the area; the divisor or factor is the line.
Oughtred gives two processes of long division. The first is identical
with the modern process, except that the divisor is written below every
remainder, each digit of the divisor being crossed out as soon as it has
been used in the partial multiplication. The second method of long
division is one of the several types of the old “scratch method.” This
antiquated process held its place by the side of the modern method in all
editions of the Clavis. The author divides 467023 by 357|0926425, giving
the following instructions: “Take as many of the first Figures of the
Divisor as are necessary, for the first Divisor, and then in every
following particular Division drop one of the Figures of the Divisor
towards the Left Hand, till you have got a competent Quotient.” He does
not explain abbreviated division as thoroughly as abbreviated
multiplication.
17
3̸0̸3̸
2̸8̸0̸3̸
1̸0̸9̸9̸3̸0̸
3̣5̣7̣|0̣9̣2̣6425) 4̸6̸7̸0̸2̸3̸ (1307|80
3̸5̸7̸0̸9̸3̸
1̸0̸7̸1̸2̸7̸
2̸5̸0̸0̸
2̸8̸6̸
Oughtred does not examine the degree of reliability or accuracy of his
processes of abbreviated multiplication and division. Here as in other
places he gives in condensed statement the mode of procedure, without
further discussion.
He does not attempt to establish the rules for the addition, subtraction,
multiplication, and division of positive and negative numbers. “If the
Signs are both alike, the Product will be affirmative, if unlike,
negative”; then he proceeds to applications. This attitude is superior to
that of many writers of the eighteenth and nineteenth centuries, on
pedagogical as well as logical grounds: pedagogically, because the
beginner in the study of algebra is not in a position to appreciate an
abstract train of thought, as every teacher well knows, and derives
better intellectual exercise from the applications of the rules to
problems; logically, because the rule of signs in multiplication does not
admit of rigorous proof, unless some other assumption is first made which
is no less arbitrary than the rule itself. It is well known that the
proofs of the rule of signs given by eighteenth-century writers are
invalid. Somewhere they involve some surreptitious assumption. This
criticism applies even to the proof given by Laplace, which tacitly
assumes the distributive law in multiplication.
A word should be said on Oughtred’s definition of + and -. He recognizes
their double function in algebra by saying (Clavis, 1631, p. 2): “Signum
additionis, sive affirmationis, est + plus” and “Signum subductionis,
sive negationis est - minus.” They are symbols which indicate the quality
of numbers in some instances and operations of addition or subtraction in
other instances. In the 1694 edition of the Clavis, thirty-four years
after the death of Oughtred, these symbols are defined as signifying
operations only, but are actually used to signify the quality of numbers
as well. In this respect the 1694 edition marks a recrudescence.
The characteristic in the Clavis that is most striking to a modern reader
is the total absence of indexes or exponents. There is much discussion in
the leading treatises of the latter part of the sixteenth and the early
part of the seventeenth century on the theory of indexes, but the modern
exponential notation, aⁿ, is of later date. The modern notation, for
positive integral exponents, first appears in Descartes’ Géométrie, 1637;
fractional and negative exponents were first used in the modern form by
Sir Isaac Newton, in his announcement of the binomial formula, in a
letter written in 1676. This total absence of our modern exponential
notation in Oughtred’s Clavis gives it a strange aspect. Like Vieta,
Oughtred uses ordinarily the capital letters, A, B, C, . . . . to
designate given numbers; A² is written Aq, A³ is written Ac; for A⁴, A⁵,
A⁶ he has, respectively, Aqq, Aqc, Acc. Only on rare occasions, usually
when some parallelism in notation is aimed at, does he use small
letters[27] to represent numbers or magnitudes. Powers of binomials or
polynomials are marked by prefixing the capital letters Q (for square), C
(for cube), QQ (for the fourth power), QC (for the fifth power), etc.
Oughtred does not express aggregation by (). Parentheses had been used by
Girard, and by Clavius as early as 1609,[28] but did not come into
general use in mathematical language until the time of Leibniz and the
Bernoullis. Oughtred indicates aggregation by writing a colon (:) at both
ends. Thus, Q:A-E: means with him (A-E)². Similarly, √q:A+E: means
√(A+E). The two dots at the end are frequently omitted when the part
affected includes all the terms of the polynomial to the end. Thus,
C:A+B-E=.. means (A+B-E)³=.. There are still further departures from this
notation, but they occur so seldom that we incline to the interpretation
that they are simply printer’s errors. For proportion Oughtred uses the
symbol (::). The proportion a:b=c:d appears in his notation a·b::c·d.
Apparently, a proportion was not fully recognized in this day as being
the expression of an equality of ratios. That probably explains why he
did not use = here as in the notation of ordinary equations. Yet Oughtred
must have been very close to the interpretation of a proportion as an
equality; for he says in his Elementi decimi Euclidis declaratio,
“proportio, sive ratio aequalis ::” That he introduced this extra symbol
when the one for equality was sufficient is a misfortune. Simplicity
demands that no unnecessary symbols be introduced. However, Oughtred’s
symbolism is certainly superior to those which preceded. Consider the
notation of Clavius.[29] He wrote 20:60=4:x, x=12, thus: “20·60·4? fiunt
12.” The insufficiency of such a notation in the more involved
expressions frequently arising in algebra is readily seen. Hence
Oughtred’s notation (::) was early adopted by English mathematicians. It
was used by John Wallis at Oxford, by Samuel Foster at Gresham College,
by James Gregory of Edinburgh, by the translators into English of Rahn’s
algebra, and by many other early writers. Oughtred has been credited
generally with the introduction of St. Andrew’s cross × as the symbol for
multiplication in the Clavis of 1631. We have discovered that this
symbol, or rather the letter x which closely resembles it, occurs as the
sign of multiplication thirteen years earlier in an anonymous “Appendix
to the Logarithmes, shewing the practise of the Calculation of Triangles
etc.” to Edward Wright’s translation of John Napier’s Descriptio,
published in 1618.[30] Later we shall give our reasons for believing that
Oughtred is the author of that “Appendix.” The × has survived as a symbol
of multiplication.
Another symbol introduced by Oughtred and found in modern books is ~,
expressing difference; thus C~D signifies the difference between C and D,
even when D is the larger number.[31] This symbol was used by John Wallis
in 1657.[32]
Oughtred represented in symbols also certain composite expressions, as
for instance A+E=Z, A-E=X, where A is greater than E. He represented by a
symbol also each of the following: A²+E², A³+E³, A²-E², A³-E³.
Oughtred practically translated the tenth book of Euclid from its
ponderous rhetorical form into that of brief symbolism. An appeal to the
eye was a passion with Oughtred. The present writer has collected the
different mathematical symbols used by Oughtred and has found more than
one hundred and fifty of them.
The differences between the seven different editions of the Clavis lie
mainly in the special parts appended to some editions and dropped in the
latest editions. The part which originally constituted the Clavis was not
materially altered, except in two or three of the original twenty
chapters. These changes were made in the editions of 1647 and 1648. After
the first edition, great stress was laid upon the theory of indices upon
the very first page, as also in passages farther on. Of course, Oughtred
did not have our modern notation of indices or exponents, but their
theory had been a part of algebra and arithmetic for some time. Oughtred
incorporated this theory in his brief exposition of the Hindu-Arabic
notation and in his explanation of logarithms. As previously pointed out,
the last three chapters of the 1631 edition were considerably rearranged
in the later editions and combined into two chapters, so that the Clavis
proper had nineteen chapters instead of twenty in the additions after the
first. These chapters consisted of applications of algebra to geometry
and were so framed as to constitute a severe test of the student’s grip
of the subject. The very last problem deals with the division of angles
into equal parts. He derives the cubic equation upon which the trisection
depends algebraically, also the equations of the fifth degree and seventh
degree upon which the divisions of the angle into 5 and 7 equal parts
depend, respectively. The exposition was severely brief, yet accurate. He
did not believe in conducting the reader along level paths or along
slight inclines. He was a guide for mountain-climbers, and woe unto him
who lacked nerve.
Oughtred lays great stress upon expansions of powers of a binomial. He
makes use of these expansions in the solution of numerical equations. To
one who does not specialize in the history of mathematics such expansions
may create surprise, for did not Newton invent the binomial theorem after
the death of Oughtred? As a matter of fact, the expansions of positive
integral powers of a binomial were known long before Newton, not only to
seventeenth-century but even to eleventh-century mathematicians.
Oughtred’s Clavis of 1631 gave the binomial coefficients for all powers
up to and including the tenth. What Newton really accomplished was the
generalization of the binomial expansion which makes it applicable to
negative and fractional exponents and converts it into an infinite
series.
As a specimen of Oughtred’s style of writing we quote his solution of
quadratic equations, accompanied by a translation into English and into
modern mathematical symbols.
As a preliminary step[33] he lets
Z=A+E and A>E;
he lets also X=A-E. From these relations he obtains identities which, in
modern notation, are ¼Z²-AE=(½Z-E)²=¼X². Now, if we know Z and AE, we can
find ½X. Then ½(Z+X)=A, and ½(Z-X)=E, and
A=½Z+√(¼Z²-AE).
Having established these preliminaries, he proceeds thus:
Datis igitur linea inaequaliter secta Z (10), & rectangulo sub
segmentis AE (21) qui gnomon est: datur semidifferentia segmentorum ½X:
& per consequens ipsa segmenta. Nam ponatur alterutrum segmentum A:
alterum erit Z-A: Rectangulum auctem est ZA-A_q=AE. Et quia dantur Z &
AE: estque ¼Z_q-AE=¼X_q: & per 5c. 18, ½Z+½X=A: & ½Z-½X=E: Aequatio sic
resoluetur: ½Z±√_q:¼Z_q-AE:=A {maius segment/minus segment.
Itaque proposita equatione, in qua sunt tres species aequaliter in
ordine tabellae adscendentes, altissima autem species ponitur negata:
Magnitudo data coefficiens mediam speciem est linea bisecanda: &
magnitudo absoluta data, ad quam sit aequatio, est rectangulum sub
segmentis inaequalibus, sine gnomon: vt ZA-A_q=AE: in numeris autem
10l-l_q=21: Estque A, vel 1l, alterutrum segmentum inaequale. Inuenitur
autem sic:
Dimidiata coefficiens median speciem est Z/2 (5); cuius quadratum est
Z_q/4 (25): ex hoc tolle AE (21) absolutum: eritque Z_q/4-AE (4)
quadratum semidifferentiae segmentorum: latus huius quadratum (2) est
semidifferentia: quam si addas ad Z/2 (5) semissem coefficientis, sive
lineae bisecandae, erit maius segment.; sin detrahas, erit minus
segment: Dico Z/2±√_q:Z_q/4-AE:=A {maius segmentum/minus segmentum.
We translate the Latin passage, using the modern exponential notation and
parentheses, as follows:
Given therefore an unequally divided line Z (10), and a rectangle
beneath the segments AE (21) which is a gnomon. Half the difference of
the segments ½X is given, and consequently the segment itself. For, if
one of the two segments is placed equal to A, the other will be Z-A.
Moreover, the rectangle is ZA-A²=AE. And because Z and AE are given,
and there is ¼Z²-AE=¼X², and by 5c.18, ½Z+½X=A, and ½Z-½X=E, the
equation will be solved thus: ½Z±√(¼Z²-AE)=A {major segment/minor
segment.
And so an equation having been proposed in which three species (terms)
are in equally ascending powers, the highest species, moreover, being
negative, the given magnitude which constitutes the middle species is
the line to be bisected. And the given absolute magnitude to which it
is equal is the rectangle beneath the unequal segments, without gnomon.
As ZA-A²=AE, or in numbers, 10x-x²=21. And A or x is one of the two
unequal segments. It may be found thus:
The half of the middle species is Z/2 (5), its square is Z²/4 (25).
From it subtract the absolute term AE (21), and Z²/4-AE (4) will be the
square of half the difference of the segments. The square root of this,
√[(Z²/2)²-AE] (2), is half the difference. If you add it to half the
coefficient Z/2 (5), or half the line to be bisected, the longer
segment is obtained; if you subtract it, the smaller segment is
obtained. I say: Z/2±√(Z²/4-AE)=A {major segment/minor segment.
The quadratic equation Aq+ZA=AE receives similar treatment. This and the
preceding equation, ZA-Aq=AE, constitute together a solution of the
general quadratic equation, x²+ax=b, provided that E or Z are not
restricted to positive values, but admit of being either positive or
negative, a case not adequately treated by Oughtred. Imaginary numbers
and imaginary roots receive no consideration whatever.
A notation suggested by Vieta and favored by Girard made vowels stand for
unknowns and consonants for knowns. This conventionality was adopted by
Oughtred in parts of his algebra, but not throughout. Near the beginning
he used Q to designate the unknown, though usually this letter stood with
him for the “square” of the expression after it.[34]
It is of some interest that Oughtred used π/δ to signify the ratio of the
circumference to the diameter of a circle. Very probably this notation is
the forerunner of the π=3.14159 . . . . used in 1706 by William Jones.
Oughtred first used π/δ in the 1647 edition of the Clavis mathematicae.
In the 1652 edition he says, “Si in circulo sit 7.22::δ·π::113.355:erit
δ·π::2 R.P: periph.” This notation was adopted by Isaac Barrow, who used
it extensively. David Gregory[35] used π/ρ in 1697, and De Moivre[36]
used c/r about 1697, to designate the ratio of the circumference to the
radius.
We quote the description of the Clavis that was given by Oughtred’s
greatest pupil, John Wallis. It contains additional information of
interest to us. Wallis devotes chap. xv of his Treatise of Algebra,
London, 1685, pp. 67-69, to Mr. Oughtred and his Clavis, saying:
Mr. William Oughtred (our Country-man) in his Clavis Mathematicae, (or
Key of Mathematicks,) first published in the Year 1631, follows Vieta
(as he did Diophantus) in the use of the Cossick Denominations;
omitting (as he had done) the names of Sursolids, and contenting
himself with those of Square and Cube, and the Compounds of these.
But he doth abridge Vieta’s Characters or Species, using only the
letters q, c, &c. which in Vieta are expressed (at length) by Quadrate,
Cube, &c. For though when Vieta first introduced this way of Specious
Arithmetick, it was more necessary (the thing being new,) to express it
in words at length: Yet when the thing was once received in practise,
Mr. Oughtred (who affected brevity, and to deliver what he taught as
briefly as might be, and reduce all to a short view,) contented himself
with single Letters instead of those words.
Thus what Vieta would have written
A Quadrate, into B Cube,
------------------------ Equal to FG Plane,
CDE Solid,
would with him be thus expressed
A_q B_c
------- = FG.
C D E
And the better to distinguish upon the first view, what quantities were
Known, and what Unknown, he doth (usually) denote the Known to
Consonants, and the Unknown by Vowels; as Vieta (for the same reason)
had done before him.
He doth also (to very great advantage) make use of several Ligatures,
or Compendious Notes, to signify Summs, Differences, and Rectangles of
several Quantities. As for instance, Of two Quantities A (the Greater),
and E (the Lesser), the Sum he calls Z, the Difference X, the Rectangle
AE. . . . .
Which being of (almost) a constant signification with him throughout,
do save a great circumlocution of words, (each Letter serving instead
of a Definition;) and are also made use of (with very great advantage)
to discover the true nature of divers intricate Operations, arising
from the various compositions of such Parts, Sums, Differences, and
Rectangles; (of which there is great plenty in his Clavis, Cap. 11, 16,
18, 19. and elsewhere,) which without such Ligatures, or Compendious
Notes, would not be easily discovered or apprehended. . . . .
I know there are who find fault with his Clavis, as too obscure,
because so short, but without cause; for his words be always full, but
not Redundant, and need only a little attention in the Reader to weigh
the force of every word, and the Syntax of it; . . . . And this, when
once apprehended, is much more easily retained, than if it were
expressed with the prolixity of some other Writers; where a Reader must
first be at the pains to weed out a great deal of superfluous Language,
that he may have a short prospect of what is material; which is here
contracted for him in a short Synopsis. . . . .
Mr. Oughtred in his Clavis, contents himself (for the most part) with
the solution of Quadratick Equations, without proceeding (or very
sparingly) to Cubick Equations, and those of Higher Powers; having
designed that Work for an Introduction into Algebra so far, leaving the
Discussion of Superior Equations for another work. . . . . He contents
himself likewise in Resolving Equations, to take notice of the
Affirmative or Positive Roots; omitting the Negative or Ablative Roots,
and such as are called Imaginary or Impossible Roots. And of those
which, he calls Ambiguous Equations, (as having more Affirmative Roots
than one,) he doth not (that I remember) any where take notice of more
than Two Affirmative Roots: (Because in Quadratick Equations, which are
those he handleth, there are indeed no more.) Whereas yet in Cubick
Equations, there may be Three, and in those of Higher Powers, yet more.
Which Vieta was well aware of, and mentioneth in some of his Writings;
and of which Mr. Oughtred could not be ignorant.
“CIRCLES OF PROPORTION” AND “TRIGONOMETRIE”
Oughtred wrote and had published three important mathematical books, the
Clavis, the Circles of Proportion,[37] and a Trigonometrie.[38] This last
appeared in the year 1657 at London, in both Latin and English.
It is claimed that the trigonometry was “neither finished nor published
by himself, but collected out of his scattered papers; and though he
connived at the printing it, yet imperfectly done, as appears by his
MSS.; and one of the printed Books, corrected by his own Hand.”[39]
Doubtless more accurate on this point is a letter of Richard Stokes who
saw the book through the press:
I have procured your Trigonometry to be written over in a fair hand,
which when finished I will send to you, to know if it be according to
your mind; for I intend (since you were pleased to give your assent) to
endeavour to print it with Mr. Briggs his Tables, and so soon as I can
get the Prutenic Tables I will turn those of the sun and moon, and send
them to you.[40]
In the preface to the Latin edition Stokes writes:
Since this trigonometry was written for private use without the
intention of having it published, it pleased the Reverend Author,
before allowing it to go to press, to expunge some things, to change
other things and even to make some additions and insert more lucid
methods of exposition.
This much is certain, the Trigonometry bears the impress characteristic
of Oughtred. Like all his mathematical writings, the book was very
condensed. Aside from the tables, the text covered only 36 pages. Plane
and spherical triangles were taken up together. The treatise is known in
the history of trigonometry as among the very earliest works to adopt a
condensed symbolism so that equations involving trigonometric functions
could be easily taken in by the eye. In the work of 1657, contractions
are given as follows: s=sine, t=tangent, se=secant, s co=cosine (sine
complement), t co=cotangent, se co=cosecant, log=logarithm, Z cru=sum of
the sides of a rectangle or right angle, X cru=difference of these sides.
It has been generally overlooked by historians that Oughtred used the
abbreviations of trigonometric functions, named above, a quarter of a
century earlier, in his Circles of Proportion, 1632, 1633. Moreover, he
used sometimes also the abbreviations which are current at the present
time, namely sin=sine, tan=tangent, sec=secant. We know that the Circles
of Proportion existed in manuscript many years before they were
published. The symbol sv for sinus versus occurs in the Clavis of 1631.
The great importance of well-chosen symbols needs no emphasis to readers
of the present day. With reference to Oughtred’s trigonometric symbols.
Augustus De Morgan said:
This is so very important a step, simple as it is, that Euler is justly
held to have greatly advanced trigonometry by its introduction. Nobody
that we know of has noticed that Oughtred was master of the
improvement, and willing to have taught it, if people would have
learnt.[41]
We find, however, that even Oughtred cannot be given the whole credit in
this matter. By or before 1631 several other writers used abbreviations
of the trigonometric functions. As early as 1624 the contractions sin for
sine and tan for tangent appear on the drawing representing Gunter’s
scale, but Gunter did not use them in his books, except in the drawing of
his scale.[42] A closer competitor for the honor of first using these
trigonometric abbreviations is Richard Norwood in his Trigonometrie,
London, 1631, where s stands for sine, t for tangent, sc for sine
complement (cosine), tc for tangent complement (cotangent), and sec for
secant. Norwood was a teacher of mathematics in London and a well-known
writer of books on navigation. Aside from the abbreviations just cited
Norwood did not use nearly as much symbolism in his mathematics as did
Oughtred.
Mention should be made of trigonometric symbols used even earlier than
any of the preceding, in “An Appendix to the Logarithmes, shewing the
practise of the Calculation of Triangles, etc.,” printed in Edward
Wright’s edition of Napier’s A Description of the Admirable Table of
Logarithmes, London, 1618. We referred to this “Appendix” in tracing the
origin of the sign ×. It contains, on p. 4, the following passage: “For
the Logarithme of an arch or an angle I set before (s), for the
antilogarithme or compliment thereof (s*) and for the Differential (t).”
In further explanation of this rather unsatisfactory passage, the author
(Oughtred?) says, “As for example: sB+BC=CA. that is, the Logarithme of
an angle B. at the Base of a plane right-angled triangle, increased by
the addition of the Logarithm of BC, the hypothenuse thereof, is equall
to the Logarithme of CA the cathetus.”
Here “logarithme of an angle B” evidently means “log sin B,” just as with
Napier, “Logarithms of the arcs” signifies really “Logarithms of the
sines of the angles.” In Napier’s table, the numbers in the column marked
“Differentiae” signify log sine minus log cosine of an angle; that is,
the logarithms of the tangents. This explains the contraction (t) in the
“Appendix.” The conclusion of all this is that as early as 1618 the signs
s, s*, t were used for sine, cosine, and tangent, respectively.
John Speidell, in his Breefe Treatise of Sphaericall Triangles, London,
1627, uses Si. for sine, T. and Tan for tangent, Se. for secant, Si. Co.
for cosine, Se. Co. for cosecant, T. Co. for cotangent.
The innovation of designating the sides and angles of a triangle by A, B,
C, and a, b, c, so that A was opposite a, B opposite b, and C opposite c,
is attributed to Leonard Euler (1753), but was first used by Richard
Rawlinson of Queen’s College, Oxford, sometimes after 1655 and before
1668. Oughtred did not use Rawlinson’s notation.[43]
In trigonometry English writers of the first half of the seventeenth
century used contractions more freely than their continental
contemporaries; even more freely, indeed, than English writers of a later
period. Von Braunmühl, the great historian of trigonometry, gives
Oughtred much praise for his trigonometry, and points out that half a
century later the army of writers on trigonometry had hardly yet reached
the standard set by Oughtred’s analysis.[44] Oughtred must be credited
also with the first complete proof that was given to the first two of
“Napier’s analogies.” His trigonometry contains seven-place tables of
sines, tangents, and secants, and six-place tables of logarithmic sines
and tangents; also seven-place logarithmic tables of numbers. At the time
of Oughtred there was some agitation in favor of a wider introduction of
decimal systems. This movement is reflected in those tables which contain
the centesimal division of the degree, a practice which is urged for
general adoption in our own day, particularly by the French.
SOLUTION OF NUMERICAL EQUATIONS
In the solution of numerical equations Oughtred does not mention the
sources from which he drew, but the method is substantially that of the
great French algebraist Vieta, as explained in a publication which
appeared in 1600 in Paris under the title, De numerosa potestatum purarum
atque adfectarum ad exegesin resolutione tractatus. In view of the fact
that Vieta’s process has been described inaccurately by leading modern
historians including H. Hankel[45] and M. Cantor,[46] it may be worth
while to go into some detail.[47] By them it is made to appear as
identical with the procedure given later by Newton. The two are not the
same. The difference lies in the divisor used. What is now called
“Newton’s method” is Newton’s method as modified by Joseph Raphson.[48]
The Newton-Raphson method of approximation to the roots of an equation
f(x)=0 is usually given the form a-[f(a)/f´(a)], where a is an
approximate value of the required root. It will be seen that the divisor
is f´(a). Vieta’s divisor is different; it is
|f(a+s₁)-f(a)|-s₁ⁿ,
where f(x) is the left of the equation f(x)=k, n is the degree of
equation, and s₁ is a unit of the denomination of the digit next to be
found. Thus in x³+420000x=247651713, it can be shown that 417 is
approximately a root; suppose that a has been taken to be 400, then
s₁=10; but if, at the next step of approximation, a is taken to be 410,
then s₁=1. In this example, taking a=400, Vieta’s divisor would have been
9120000; Newton’s divisor would have been 900000.
A comparison of Vieta’s method with the Newton-Raphson method reveals the
fact that Vieta’s divisor is more reliable, but labors under the very
great disadvantage of requiring a much larger amount of computation. The
latter divisor is accurate enough and easier to compute. Altogether the
Newton-Raphson process marks a decided advance over that of Vieta.
As already stated, it is the method of Vieta that Oughtred explains. The
Englishman’s exposition is an improvement on that of Vieta, printed forty
years earlier. Nevertheless, Oughtred’s explanation is far from easy to
follow. The theory of equations was at that time still in its primitive
stage of development. Algebraic notation was not sufficiently developed
to enable the argument to be condensed into a form easily surveyed. So
complicated does Vieta’s process of approximation appear that M. Cantor
failed to recognize that Vieta possessed a uniform mode of procedure. But
when one has in mind the general expression for Vieta’s divisor which we
gave above, one will recognize that there was marked uniformity in
Vieta’s approximations.
Oughtred allows himself twenty-eight sections in which to explain the
process and at the close cannot forbear remarking that 28 is a “perfect”
number (being equal to the sum of its divisors, 1, 2, 4, 7, 14).
The early part of his exposition shows how an equation may be transformed
so as to make its roots 10, 100, 1000, or 10^m times smaller. This
simplifies the task of “locating a root”; that is, of finding between
what integers the root lies.
Taking one of Oughtred’s equations, x⁴-72x³+238600x=8725815, upon
dividing 72x³ by 10, 238600x by 1000, and 8725815 by 10,000, we obtain
x⁴-7·2x³+238·6x=872·5. Dividing both sides by x, we obtain
x³+238·6-7·2x²=x)872·5. Letting x=4, we have 64+238·6-115·2=187·4.
But 4)872·5(218·1; 4 is too small. Next let x=5, we have
125+238·6-180=183·6.
But 5)872·5(174·5; 5 is too large. We take the lesser value, x=4, or in
the original equation, x=40. This method may be used to find the second
digit in the root. Oughtred divides both sides of the equation by x², and
obtains x²+x)238600-72x=x²)8725815. He tries x=47 and x=48, and finds
that x=47.
He explains also how the last computation may be done by logarithms.
Thereby he established for himself the record of being the first to use
logarithms in the solution of affected equations.
As an illustration of Oughtred’s method of approximation after the root
sought has been located, we have chosen for brevity a cubic in preference
to a quartic. We selected the equation x³+420000x=247651713. By the
process explained above a root is found to lie between x=400 and x=500.
From this point on, the approximation as given by Oughtred is as shown on
p. 43.
In further explanation of this process, observe that the given equation
is of the form L_c+C_qL=D_c, where L_c is our x, C_q=420000,
D_c=247651713. In the first step of approximation, let L=A+E, where A=400
and E is, as yet, undetermined. We have
L_c=(A+E)³=A³+3A²E+3AE²+E³
and
C_qL=420000(A+E).
Subtract from 247651713 the sum of the known terms A³ (his A_c) and
420000 A (his C_qA). This sum is 232000000 the remainder is 15651713.
“Exemplum II
1c+42̣00̣00̣l=247̇651̇7̣1̣3̣̇
Hoc est, L_c+C_qL=D_c
2 4 7̇ | 6 5 1̇ | 7̣ 1̣ 3̣̇ | ( 4 1 7
------+-------+-------+------------
4 2 | 0 0 0 | 0 | C_q
------+-------+-------+------------
6 4 | | | A_c
1 6 8 | 0 0 0 | 0 | C_q A
------+-------+-------+------------
2 3 2 | 0 0 0 | 0 | Ablatit.
===================================
R 1 5 | 6 5 1̇ | 7 1 3̣ |
------+-------+-------+------------
4 | 8 | | 3 A_q
| 1 2 | | 3 A
4 | 2 0 0 | 0 0 | C_q
------+-------+-------+------------
9 | 1 2 0 | 0 0 | Divisor.
------+-------+-------+------------
4 | 8 | | 3 A_q E
| 1 2 | | 3 A E_q
| 1 | | E_c
4 | 2 0 0 | 0 0 | C_q E
------+-------+-------+------------
9 | 1 2 1 | 0 0 | Ablatit.
===================================
R 6 | 5 3 0 | 7 1 3̣̇ | 4 | 1 |
------+-------+-------+------------ ----+-----+---
| 5 0 4 | 3 | 3 A_q | |
| 1 | 2 3 | 3 A 1 6 | 8 |
| 4 2 0 | 0 0 0 | C_q | 1 |
------+-------+-------+------------ ----+-----+---
| 9 2 5 | 5 3 0 | Divisor. 1 6 8 1
------+-------+-------+------------
3 | 5 3 0 | 1 | 3 A_q E
| 6 0 | 2 7 | 3 A E_q
| | 3 4 3 | E_c
2 | 9 4 0 | 0 0 0 | C_q E
------+-------+-------+------------
6 | 5 3 0 | 7 1 3 | Ablatit.”
Next, he evaluates the coefficients of E in 3A²E and 420000E, also 3A,
the coefficient of E². He obtains 3A²=480000, 3A=1200, C_q=420000. He
interprets 3A² and C_q as tens, 3A as hundreds. Accordingly, he obtains
as their sum 9120000, which is the divisor for finding the second digit
in the approximation. Observe that this divisor is the value of
|f(a+s₁)-f(a)|-s₁ⁿ in our general expression, where a=400, s₁=10, n=3,
f(x)=x³+420000x.
Dividing the remainder 15651713 by 9120000, he obtains the integer 1 in
ten’s place; thus E=10, approximately. He now computes the terms 3A²E,
3AE² and E³ to be, respectively, 4800000, 120000, 1000. Their sum is
9121000. Subtracting it from the previous remainder, 15651713, leaves the
new remainder, 6530713.
From here on each step is a repetition of the preceding step. The new A
is 410, the new E is to be determined. We have now in closer
approximation, L=A+E. This time we do not subtract A³ and C_qA, because
this subtraction is already affected by the preceding work.
We find the second trial divisor by computing the sum of 3A², 3A and C_q;
that is, the sum of 504300, 1230, 420000, which is 925530. Again, this
divisor can be computed by our general expression for divisors, by taking
a=410, s₁=1, n=3.
Dividing 6530713 by 925530 yields the integer 7. Thus E=7. Computing
3A²E, 3AE², E³ and subtracting their sum, the remainder is 0. Hence 417
is an exact root of the given equation.
Since the extraction of a cube root is merely the solution of a pure
cubic equation, x³=n, the process given above may be utilized in finding
cube roots. This is precisely what Oughtred does in chap. xiv of his
Clavis. If the foregoing computation is modified by putting C_q=0, the
process will yield the approximate cube root of 247651713.
Oughtred solves 16 examples by the process of approximation here
explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In
all cases he finds only one or two real roots. Of the roots sought, five
are irrational, the remaining are rational and are computed to their
exact values. Three of the computed roots have 2 figures each, 9 roots
have 3 figures each, 4 roots have 4 figures each. While no attempt is
made to secure all the roots—methods of computing complex roots were
invented much later—he computes roots of equations which involve large
coefficients and some of them are of a degree as high as the fifth. In
view of the fact that many editions of the Clavis were issued, one
impression as late as 1702, it contributed probably more than any other
book to the popularization of Vieta’s method in England.
Before Oughtred, Thomas Harriot and William Milbourn are the only
Englishmen known to have solved numerical equations of higher degrees.
Milbourn published nothing. Harriot slightly modified Vieta’s process by
simplifying somewhat the formation of the trial divisor. This method of
approximation was the best in existence in Europe until the publication
by Wallis in 1685 of Newton’s method of approximation.
It should be stated that, before the time of Newton, the best method of
approximation to the roots of numerical equations existed, not in Europe,
but in China. As early as the thirteenth century the Chinese possessed a
method which is almost identical with what is known today as “Horner’s
method.”
LOGARITHMS
Oughtred’s treatment of logarithms is quite in accordance with the more
recent practice.[49] He explains the finding of the “index” (our
“characteristic”); he states that “the sum of two Logarithms is the
Logarithm of the Product of their Valors; and their difference is the
Logarithm of the Quotient,” that “the Logarithm of the side [436] drawn
upon the Index number [2] of dimensions of any Potestas is the logarithm
of the same Potestas” [436²], that “the logarithm of any Potestas [436²]
divided by the number of its dimensions [2] affordeth the Logarithm of
its Root [436].” These statements of Oughtred occur for the first time in
the Key of the Mathematicks of 1647; the Clavis of 1631 contains no
treatment of logarithms.
If the characteristic of a logarithm is negative, Oughtred indicates this
fact by placing the - above the characteristic. He separates the
characteristic and mantissa by a comma, but still uses the sign |_ to
indicate decimal fractions. He uses the contraction “log.”
INVENTION OF THE SLIDE RULE; CONTROVERSY ON PRIORITY OF INVENTION
Oughtred’s most original line of scientific activity is the one least
known to the present generation. Augustus De Morgan, in speaking of
Oughtred, who was sometimes called “Oughtred Aetonensis,” remarks: “He is
an animal of extinct race, an Eton mathematician. Few Eton men, even of
the minority which knows what a sliding rule is, are aware that the
inventor was of their own school and college.”[50] The invention of the
slide rule has, until recently,[51] been a matter of dispute; it has been
erroneously ascribed to Edmund Gunter, Edmund Wingate, Seth Partridge,
and others. We have been able to establish that William Oughtred was the
first inventor of slide rules, though not the first to publish thereon.
We shall see that Oughtred invented slide rules about 1622, but the
descriptions of his instruments were not put into print before 1632 and
1633. Meanwhile one of his own pupils, Richard Delamain, who probably
invented the circular slide rule independently, published a description
in 1630, at London, in a pamphlet of 32 pages entitled Grammelogia; or
the Mathematicall Ring. In editions of this pamphlet which appeared
during the following three or four years, various parts were added on,
and some parts of the first and second editions eliminated. Thus Delamain
antedates Oughtred two years in the publication of a description of a
circular slide rule. But Oughtred had invented also a rectilinear slide
rule, a description of which appeared in 1633. To the invention of this
Oughtred has a clear title. A bitter controversy sprang up between
Delamain on one hand, and Oughtred and some of his pupils on the other,
on the priority and independence of invention of the circular slide rule.
Few inventors and scientific men are so fortunate as to escape contests.
The reader needs only to recall the disputes which have arisen, involving
the researches of Sir Isaac Newton and Leibniz on the differential and
integral calculus, of Thomas Harriot and René Descartes relating to the
theory of equations, of Robert Mayer, Hermann von Helmholtz, and Joule on
the principle of the conservation of energy, or of Robert Morse, Joseph
Henry, Gauss and Weber, and others on the telegraph, to see that
questions of priority and independence are not uncommon. The controversy
between Oughtred and Delamain embittered Oughtred’s life for many years.
He refers to it in print on more than one occasion. We shall confine
ourselves at present to the statement that it is by no means clear that
Delamain stole the invention from Oughtred; Delamain was probably an
independent inventor. Moreover, it is highly probable that the
controversy would never have arisen, had not some of Oughtred’s pupils
urged and forced him into it. William Forster stated in the preface to
the Circles of Proportion of 1632 that while he had been carefully
preparing the manuscript for the press, “another to whom the Author
[Oughtred] in a louing confidence discouered this intent, using more hast
then good speed, went about to preocupate.” It was this passage which
started the conflagration. Another pupil, W. Robinson, wrote to Oughtred,
when the latter was preparing his Apologeticall Epistle as a reply to
Delamain’s countercharges: “Good sir, let me be beholden to you for your
Apology whensoever it comes forth, and (if I speak not too late) let me
entreat you, whip ignorance well on the blind side, and we may turn him
round, and see what part of him is free.”[52] As stated previously,
Oughtred’s circular slide rule was described by him in his Circles of
Proportion, London, 1632, which was translated from Oughtred’s Latin
manuscript and then seen through the press by his pupil, William Forster.
In 1633 appeared An Addition vnto the Vse of the Instrvment called the
Circles of Proportion which contained at the end “The Declaration of the
two Rulers for Calculation,” giving a description of Oughtred’s
rectilinear slide rule. This Addition was bound with the Circles of
Proportion as one volume. About the same time Oughtred described a
modified form of the rectilinear slide rule, to be used in London for
gauging.[53]