OUGHTRED’S IDEAS ON THE TEACHING OF MATHEMATICS
GENERAL STATEMENT
Nowhere has Oughtred given a full and systematic exposition of his views
on mathematical teaching. Nevertheless, he had very pronounced and
clear-cut ideas on the subject. That a man who was not a teacher by
profession should have mature views on teaching is most interesting. We
gather his ideas from the quality of the books he published, from his
prefaces, and from passages in his controversial writing against
Delamain. As we proceed to give quotations unfolding Oughtred’s views, we
shall observe that three points receive special emphasis: (1) an appeal
to the eye through suitable symbolism; (2) emphasis upon rigorous
thinking; (3) the postponement of the use of mathematical instruments
until after the logical foundations of a subject have been thoroughly
mastered.
The importance of these tenets is immensely reinforced by the conditions
of the hour. This voice from the past speaks wisdom to specialists of
today. Recent methods of determining educational values and the modern
cult of utilitarianism have led some experts to extraordinary
conclusions. Laboratory methods of testing, by the narrowness of their
range, often mislead. Thus far they have been inferior to the word of a
man of experience, insight, and conviction.
MATHEMATICS, “A SCIENCE OF THE EYE”
Oughtred was a great admirer of the Greek mathematicians—Euclid,
Archimedes, Apollonius of Perga, Diophantus. But in reading their works
he experienced keenly what many modern readers have felt, namely, that
the almost total absence of mathematical symbols renders their writings
unnecessarily difficult to read. Statements that can be compressed into a
few well-chosen symbols which the eye is able to survey as a whole are
expressed in long-drawn-out sentences. A striking illustration of the
importance of symbolism is afforded by the history of the formula
ix=log(cos x+i sin x).
It was given in Roger Cotes’ Harmonia mensurarum, 1722, not in symbols,
but expressed in rhetorical form, destitute of special aids to the eye.
The result was that the theorem remained in the book undetected for 185
years and was meanwhile rediscovered by others. Owing to the prominence
of Cotes as a mathematician it is very improbable that such a thing could
have happened had the theorem been thrust into view by the aid of
mathematical symbols.
In studying the ancient authors Oughtred is reported to have written down
on the margin of the printed page some of the theorems and their proofs,
expressed in the symbolic language of algebra.
In the preface of his Clavis of 1631 and of 1647 he says:
Wherefore, that I might more clearly behold the things themselves, I
uncasing the Propositions and Demonstrations out of their covert of
words, designed them in notes and species appearing to the very eye.
After that by comparing the divers affections of Theorems, inequality,
proportion, affinity, and dependence, I tryed to educe new out of them.
It was this motive which led him to introduce the many abbreviations in
algebra and trigonometry to which reference has been made in previous
pages. The pedagogical experience of recent centuries has indorsed
Oughtred’s view, provided of course that the pupil is carefully taught
the exact meaning of the symbols. There have been and there still are
those who oppose the intensive use of symbolism. In our day the new
symbolism for all mathematics, suggested by the school of Peano in Italy,
can hardly be said to be received with enthusiasm. In Oughtred’s day
symbolism was not yet the fashion. To be convinced of this fact one need
only open a book of Edmund Gunter, with whom Oughtred came in contact in
his youth, or consult the Principia of Sir Isaac Newton, who flourished
after Oughtred. The mathematical works of Gunter and Newton, particularly
the former, are surprisingly destitute of mathematical symbols. The
philosopher Hobbes, in a controversy with John Wallis, criticized the
latter for that “Scab of Symbols,” whereupon Wallis replied:
I wonder how you durst touch M. Oughtred for fear of catching the Scab.
For, doubtlesse, his book is as much covered over with the Scab of
Symbols, as any of mine. . . . . As for my Treatise of Conick Sections,
you say, it is covered over with the Scab of Symbols, that you had not
the patience to examine whether it is well or ill demonstrated.[142]
Oughtred maintained his view of the importance of symbols on many
different occasions. Thus, in his Circles of Proportion, 1632, p. 20:
This manner of setting downe Theoremes, whether they be Proportions, or
Equations, by Symboles or notes of words, is most excellent,
artificiall, and doctrinall. Wherefore I earnestly exhort every one,
that desireth though but to looke into these noble Sciences
Mathematicall, to accustome themselves unto it: and indeede it is
easie, being most agreeable to reason, yea even to sence. And out of
this working may many singular consectaries be drawne: which without
this would, it may be, for ever lye hid.
RIGOROUS THINKING AND THE USE OF INSTRUMENTS
The author’s elevated concept of mathematical study as conducive to
rigorous thinking shines through the following extract from his preface
to the 1647 Clavis:
. . . . Which Treatise being not written in the usuall synthetical
manner, nor with verbous expressions, but in the inventive way of
Analitice, and with symboles or notes of things instead of words,
seemed unto many very hard; though indeed it was but their owne
diffidence, being scared by the newnesse of the delivery; and not any
difficulty in the thing it selfe. For this specious and symbolicall
manner, neither racketh the memory with multiplicity of words, nor
chargeth the phantasie with comparing and laying things together; but
plainly presenteth to the eye the whole course and processe of every
operation and argumentation.
Now my scope and intent in the first Edition of that my Key was, and in
this New Filing, or rather forging of it, is, to reach out to the
ingenious lovers of these Sciences, as it were Ariadnes thread, to
guide them through the intricate Labyrinth of these studies, and to
direct them for the more easie and full understanding of the best and
antientest Authors. . . . . That they may not only learn their
propositions, which is the highest point of Art that most Students aime
at; but also may perceive with what solertiousnesse, by what engines of
aequations, Interpretations, Comparations, Reductions, and
Disquisitions, those antient Worthies have beautified, enlarged, and
first found out this most excellent Science. . . . . Lastly, by framing
like questions problematically, and in a way of Analysis, as if they
were already done, resolving them into their principles, I sought out
reasons and means whereby they might be effected. And by this course of
practice, not without long time, and much industry, I found out this
way for the helpe and facilitation of Art.
Still greater emphasis upon rigorous thinking in mathematics is laid in
the preface to the Circles of Proportion and in some parts of his
Apologeticall Epistle against Delamain. In that preface William Forster
quotes the reply of Oughtred to the question how he (Oughtred) had for so
many years concealed his invention of the slide rule from himself
(Forster) whom he had taught so many other things. The reply was:
That the true way of Art is not by Instruments, but by Demonstration:
and that it is a preposterous course of vulgar Teachers, to begin with
Instruments, and not with the Sciences, and so in-stead of Artists, to
make their Scholers only doers of tricks, and as it were Iuglers: to
the despite of Art, losse of previous time, and betraying of willing
and industrious wits, vnto ignorance, and idlenesse. That the vse of
Instruments is indeed excellent, if a man be an Artist: but
contemptible, being set and opposed to Art. And lastly, that he meant
to commend to me, the skill of Instruments, but first he would haue me
well instructed in the Sciences.”
Delamain took a different view, arguing that instruments might very well
be placed in the hands of pupils from the start. At the time of this
controversy Delamain supported himself by teaching mathematics in London
and he advertised his ability to give instruction in mathematics,
including the use of instruments. Delamain brought the charge against
Oughtred of unjustly calling “many of the [British] Nobility and Gentry
doers of trickes and juglers.” To this Oughtred replies:
As I did to Delamain and to some others, so I did to William Forster: I
freely gave him my helpe and instruction in these faculties: only this
was the difference, I had the very first moulding (as I may say) of
this latter: But Delamain was already corrupted with doring upon
Instruments, and quite lost from ever being made an Artist: I suffered
not William Forster for some time so much as speake of any Instrument,
except only the Globe it selfe; and to explicate, and worke the
questions of the Sphaere, by the way of the Analemma: which also
himselfe did describe for the present occasion. And this my restraint
from such pleasing avocations, and holding him to the strictnesse of
percept, brought forth this fruit, that in short time, even by his owne
skill, he could not onely use any Instrument he should see, but also
was able to delineate the like, and devise others.[143]
As representing Delamain’s views, we make the following selection from
his Grammelogia (London, about 1633), the part near the end of the book
and bearing the title, “In the behalfe of vulgar Teachers and others,”
where Delamain refers to Oughtred’s charge that the scholars of “vulgar”
teachers are “doers of tricks, as it were iuglers.” Delamain says:
. . . . Which words are neither cautelous, nor subterfugious, but are
as downe right in their plainnesse, as they are touching, and
pernitious, by two much derogating from many, and glancing upon many
noble personages, with too grosse, if not too base an attribute, in
tearming them doers of tricks, as it were to iuggle: because they
perhaps make use of a necessitie in the furnishing of themselves with
such knowledge by Practicall Instrumentall operation, when their more
weighty negotiations will not permit them for Theoreticall figurative
demonstration; those that are guilty of the aspertion, and are touched
therewith may answer for themselves, and studie to be more
Theoreticall, than Practicall: for the Theory, is as the Mother that
produceth the daughter, the very sinewes and life of Practise, the
excellencie and highest degree of true Mathematicall Knowledge: but for
those that would make but a step as it were into that kind of Learning,
whose onely desire is expedition, and facilitie, both which by the
generall consent of all are best effected with Instrument, rather then
with tedious regular demonstrations, it was ill to checke them so
grosly, not onely in what they have Practised, but abridging them also
of their liberties with what they may Practise, which aspertion may not
easily be slighted off by any glosse or Apologie, without an Ingenuous
confession, or some mentall reservation: To which vilification,
howsoever, in the behalfe of my selfe, and others, I answer; That
Instrumentall operation is not only the Compendiating, and facilitating
of Art, but even the glory of it, whole demonstration both of the
making, and operation is soly in the science, and to an Artist or
disputant proper to be knowne, and so to all, who would truly know the
cause of the Mathematicall operations in their originall; But, for none
to know the use of a Mathematicall Instrumen[t], except he knowes the
cause of its operation, is somewhat too strict, which would keepe many
from affecting the Art, which of themselves are ready enough every
where, to conceive more harshly of the difficultie, and impossibilitie
of attayning any skill therein, then it deserves, because they see
nothing but obscure propositions, and perplex and intricate
demonstrations before their eyes, whose unsavoury tartnes, to an
unexperienced palate like bitter pills is sweetned over, and made
pleasant with an Instrumentall compendious facilitie, and made to goe
downe the more readily, and yet to retaine the same vertue, and
working; And me thinkes in this queasy age, all helpes may bee used to
procure a stomacke, all bates and invitations to the declining studie
of so noble a Science, rather then by rigid Method and generall Lawes
to scarre men away. All are not of like disposition, neither all (as
was sayd before) propose the same end, some resolve to wade, others to
put a finger in onely, or wet a hand: now thus to tye them to an
obscure and Theoricall forme of teaching, is to crop their hope, even
in the very bud. . . . . The beginning of a mans knowledge even in the
use of an Instrument, is first founded on doctrinal precepts, and these
precepts may be conceived all along in its use: and are so farre from
being excluded, that they doe necessarily concomitate and are contained
therein: the practicke being better understood by the doctrinall part,
and this later explained by the Instrumentall, making precepts obvious
unto sense, and the Theory going along with the Instrument, better
informing and inlightning the understanding, etc. vis vnita fortior, so
as if that in Phylosophy bee true, Nihil est [in] intellectu quod non
prius fuit in sensu.
The difference between Oughtred and Delamain as to the use of
mathematical instruments raises important questions. Should the slide
rule be placed in the hands of a boy before, or after, he has mastered
the theory of logarithms? Should logarithmic tables be withheld from him
until the theoretical foundation is laid in the mind of the pupil? Is it
a good thing to let a boy use a surveying instrument unless he first
learns trigonometry? Is it advisable to permit a boy to familiarize
himself with the running of a dynamo before he has mastered the
underlying principles of electricity? Does the use of instruments
ordinarily discourage a boy from mastery of the theory? Or does such
manipulation constitute a natural and pleasing approach to the abstract?
On this particular point, who showed the profounder psychological
insight, Oughtred or Delamain?
In July, 1914, there was held in Edinburgh a celebration of the
three-hundredth anniversary of the invention of logarithms. On that
occasion there was collected at Edinburgh university one of the largest
exhibits ever seen of modern instruments of calculation. The opinion was
expressed by an experienced teacher that “weapons as those exhibited
there are for men and not for boys, and such danger as there may be in
them is of the same character as any form of too early specialization.”
It is somewhat of a paradox that Oughtred, who in his student days and
during his active years felt himself impelled to invent sun-dials,
planispheres, and various types of slide rules—instruments which
represent the most original contributions which he handed down to
posterity—should discourage the use of such instruments in teaching
mathematics to beginners. That without the aid of instruments he himself
should have succeeded so well in attracting and inspiring young men
constitutes the strongest evidence of his transcendent teaching ability.
It may be argued that his pedagogic dogma, otherwise so excellent, here
goes contrary to the course he himself followed instinctively in his
self-education along mathematical lines. We read that Sir Isaac Newton,
as a child, constructed sun-dials, windmills, kites, paper lanterns, and
a wooden clock. Should these activities have been suppressed? Ordinary
children are simply Isaac Newtons on a smaller intellectual scale. Should
their activities along these lines be encouraged or checked?
On the other hand, it may be argued that the paradox alluded to above
admits of explanation, like all paradoxes, and that there is no
inconsistency between Oughtred’s pedagogic views and his own course of
development. If he invented sun-dials, he must have had a comprehension
of the cosmic motions involved; if he solved spherical triangles
graphically by the aid of the planisphere, he must have understood the
geometry of the sphere, so far as it relates to such triangles; if he
invented slide rules, he had beforehand a thorough grasp of logarithms.
The question at issue does not involve so much the invention of
instruments, as the use by the pupil of instruments already constructed,
before he fully understands the theory which is involved. Nor does Sir
Isaac Newton’s activity as a child establish Delamain’s contention. Of
course, a child should not be discouraged from manual activity along the
line of producing interesting toys in imitation of structures and
machines that he sees, but to introduce him to the realm of abstract
thought by the aid of instruments is a different proposition, fraught
with danger. A boy may learn to use a slide rule mechanically and,
because of his ability to obtain practical results, feel justified in
foregoing the mastery of underlying theory; or he may consider the
ability of manipulating a surveying instrument quite sufficient, even
though he be ignorant of geometry and trigonometry; or he may learn how
to operate a dynamo and an electric switchboard and be altogether
satisfied, though having no grasp of electrical science. Thus instruments
draw a youth aside from the path leading to real intellectual attainments
and real efficiency; they allure him into lanes which are often blind
alleys. Such were the views of Oughtred.
Who was right, Oughtred or Delamain? It may be claimed that there is a
middle ground which more nearly represents the ideal procedure in
teaching. Shall the slide rule be placed in the student’s hands at the
time when he is engaged in the mastery of principles? Shall there be an
alternate study of the theory of logarithms and of the slide rule—on the
idea of one hand washing the other—until a mastery of both the theory and
the use of the instrument has been attained? Does this method not produce
the best and most lasting results? Is not this Delamain’s actual
contention? We leave it to the reader to settle these matters from his
own observation, knowledge, and experience.
NEWTON’S COMMENTS ON OUGHTRED
Oughtred is an author who has been found to be of increasing interest to
modern historians of mathematics. But no modern writer has, to our
knowledge, pointed out his importance in the history of the teaching of
mathematics. Yet his importance as a teacher did receive recognition in
the seventeenth century by no less distinguished a scientist than Sir
Isaac Newton. On May 25, 1694, Sir Isaac Newton wrote a long letter in
reply to a request for his recommendation on a proposed new course of
study in mathematics at Christ’s Hospital. Toward the close of his
letter, Newton says:
And now I have told you my opinion in these things, I will give you Mr.
Oughtred’s, a Man whose judgment (if any man’s) may be safely relyed
upon. For he in his book of the circles of proposition, in the end of
what he writes about Navigation (page 184) has this exhortation to
Seamen. “And if,” saith he, “the Masters of Ships and Pilots will take
the pains in the Journals of their Voyages diligently and faithfully to
set down in severall columns, not onely the Rumb they goe on and the
measure of the Ships way in degrees, and the observation of Latitude
and variation of their compass; but alsoe their conjectures and reason
of their correction they make of the aberrations they shall find, and
the qualities and condition of their ship, and the diversities and
seasons of the winds, and the secret motions or agitations of the Seas,
when they begin, and how long they continue, how farr they extend and
with what inequality; and what else they shall observe at Sea worthy
consideration, and will be pleased freely to communicate the same with
Artists, such as are indeed skilfull in the Mathematicks and lovers and
enquirers of the truth: I doubt not but that there shall be in
convenient time, brought to light many necessary precepts which may
tend to y^e perfecting of Navigation, and the help and safety of such
whose Vocations doe inforce them to commit their lives and estates in
the vast Ocean to the providence of God.” Thus farr that very good and
judicious man Mr. Oughtred. I will add, that if instead of sending the
Observations of Seamen to able Mathematicians at Land, the Land would
send able Mathematicians to Sea, it would signify much more to the
improvem^t of Navigation and safety of Mens lives and estates on that
element.[144]
May Oughtred prove as instructive to the modern reader as he did to
Newton!
Footnotes
[1]Aubrey’s Brief Lives, ed. A. Clark, Vol. II, Oxford, 1898, p. 106.
[2]“To the English Gentrie, and all others studious of the Mathematicks,
which shall bee Readers hereof. The just Apologie of Wil: Ovghtred,
against the slaunderous insimulations of Richard Delamain, in a
Pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica
logarithmorum projectio circularis” [1633?], p. 8. Hereafter we shall
refer to this pamphlet as the Apologeticall Epistle, this name
appearing on the page-headings.
[3]Companion to the [British] Almanac of 1837, p. 28, in an article by
Augustus De Morgan on “Notices of English Mathematical and
Astronomical Writers between the Norman Conquest and the Year 1600.”
[4]New and General Biographical Dictionary (John Nichols), London, 1784,
art. “Oughtred.”
[5]Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, p. 132.
[6]Skeleton Collegii Regalis Cantab.: Or A Catalogue of All the Provosts,
Fellows and Scholars, of the King’s College . . . . since the
Foundation Thereof, Vol. II, “William Oughtred.”
[7]Aubrey, op. cit., Vol. II, p. 107.
[8]Rigaud, Correspondence of Scientific Men of the Seventeenth Century,
Oxford, Vol. I, 1841, p. 5.
[9]Aubrey, op. cit., Vol. II, p. 110.
[10]Ibid., p. 111.
[11]Op. cit., Vol. II, p. 132.
[12]Mr. William Lilly’s History of His Life and Times, From the Year 1602
to 1681, London, 1715, p. 58.
[13]Rigaud, op. cit., Vol. I, p. 60.
[14]Aubrey, op. cit., Vol. II, p. 107.
[15]Rigaud, op. cit., Vol. I, p. 16.
[16]Owen Manning, op. cit., p. 132.
[17]New and General Biographical Dictionary (John Nichols), London, 1784,
art. “Oughtred.”
[18]Op. cit., Vol. II, p. 110.
[19]Rev. Owen Manning, The History and Antiquities of Surrey, Vol. II,
London, 1809, p. 132.
[20]Op. cit., Vol. II, 1898, p. 111.
[21]Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago and
London, 1915, Vol. II, p. 303.
[22]The full title of the Clavis of 1631 is as follows: Arithmeticae in
numeris et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae,
atqve adeo totivs mathematicae, qvasi clavis est.—Ad nobilissimvm
spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur,
Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae &
Svrriae, Comitis Mareschalli Angliae, &c filium.—Londini, Apud Thomam
Harpervm. M.DC.XXXI.
In all there appeared five Latin editions, the second in 1648 at
London, the third in 1652 at Oxford, the fourth in 1667 at Oxford, the
fifth in 1693 and 1698 at Oxford. There were two independent English
editions: the first in 1647 at London, translated in greater part by
Robert Wood of Lincoln College, Oxford, as is stated in the preface to
the 1652 Latin edition; the second in 1694 and 1702 is a new
translation, the preface being written and the book recommended by the
astronomer Edmund Halley. The 1694 and 1702 impressions labored under
the defect of many sense-disturbing errors due to careless reading of
the proofs. All the editions of the Clavis, after the first edition,
had one or more of the following tracts added on:
Eq.=De Aequationum affectarvm resolvtione in numeris.
Eu.=Elementi decimi Euclidis declaratio.
So.=De Solidis regularibus, tractatus.
An.=De Anatocismo, sive usura composita.
Fa.=Regula falsae positionis.
Ar.=Theorematum in libris Archimedis de Sphaera & cylindro declaratio.
Ho.=Horologia scioterica in plano, geometricè delineandi modus.
The abbreviated titles given here are, of course, our own. The lists
of tracts added to the Clavis mathematicae of 1631 in its later
editions, given in the order in which the tracts appear in each
edition, are as follows: Clavis of 1647, Eq., An., Fa., Ho.; Clavis of
1648, Eq., An., Fa., Eu., So.; Clavis of 1652, Eq., Eu., So., An.,
Fa., Ar., Ho.; Clavis of 1667, Eq., Eu., So., An., Fa., Ar., Ho.;
Clavis of 1693 and 1698, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of
1694 and 1702, Eq.
The title-page of the Clavis was considerably modified after the first
edition. Thus, the 1652 Latin edition has this title-page: Guilelmi
Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia Socii,
Clavis mathematicae denvo limata, sive potius fabricata. Cum aliis
quibusdam ejusdem commentationibus, quae in sequenti pagina
recensentur. Editio tertia auctior & emendatior. Oxoniae, Excudebat
Leon. Lichfield, Veneunt apud Tho. Robinson. 1652.
[23]Rigaud, op. cit., Vol. II, p. 476.
[24]See, for instance, the Clavis mathematicae of 1652, where he
expresses himself thus (p. 4): “Speciosa haec Arithmetica arti
Analyticae (per quam ex sumptione quaesiti, tanquam noti, investigatur
quaesitum) multo accommodatior est, quam illa numerosa.”
[25]Oughtred, The Key of the Mathematicks, London, 1647, p. 4.
[26]Clavis 1694, p. 19, and the Clavis of 1631, p. 8.
[27]See for instance, Oughtred’s Elementi decimi Euclidis declaratio,
1652, p. 1, where he uses A and E, and also a and e.
[28]See Christophori Clavii Bambergensis Operum mathematicorum, tomus
secundus, Moguntiae, M.DC.XI, algebra, p. 39.
[29]Christophori Clavii operum mathematicorum Tomus Secundus, Moguntiae,
M.DC.XI, Epitome arithmeticae, p. 36.
[30]See F. Cajori, “The Cross × as a Symbol of Multiplication,” in
Nature, Vol. XCIV (1914), p. 363.
[31]See Elementi decimi Euclidis declaratio, 1652, p. 2.
[32]See Johannis Wallisii Operum mathematicorum pars prima, Oxonii, 1657,
p. 247.
[33]Clavis of 1631, chap. xix, sec. 5, p. 50.
[34]We have noticed the representation of known quantities by consonants
and the unknown by vowels in Wingate’s Arithmetick made easie, edited
by John Kersey, London, 1650, algebra, p. 382; and in the second part,
section 19, of Jonas Moore’s Arithmetick in two parts, London, 1660,
Moore suggests as an alternative the use of z, y, x, etc., for the
unknowns. The practice of representing unknowns by vowels did not
spread widely in England.
[35]Philosophical Transactions, Vol. XIX, No. 231, London, p. 652.
[36]Ibid., Vol. XIX, p. 56.
[37]There are two title-pages to the edition of 1632. The first
title-page is as follows: The Circles of Proportion and The
Horizontall Instrument. Both invented, and the vses of both Written in
Latine by Mr. W. O. Translated into English: and set forth for the
publique benefit by William Forster. London. Printed for Elias Allen
maker of these and all other mathematical Instruments, and are to be
sold at his shop over against St. Clements church with out
Temple-barr. 1632. T. Cecill Sculp.
In 1633 there was added the following, with a separate title-page: An
addition vnto the Vse of the Instrvment called the Circles of
Proportion. . . . . London, 1633, this being followed by Oughtred’s To
the English Gentrie etc. In the British Museum there is a copy of
another impression of the Circles of Proportion, dated 1639, with the
Addition vnto the Vse of the Instrument etc., bearing the original
date, 1633, and with the epistle, To the English Gentrie, etc.,
inserted immediately after Forster’s dedication, instead of at the end
of the volume.
[38]The complete title of the English edition is as follows:
Trigonometrie, or, The manner of calculating the Sides and Angles of
Triangles, by the Mathematical Canon, demonstrated. By William
Oughtred Etonens. And published by Richard Stokes Fellow of Kings
Colledge in Cambridge, and Arthur Haughton Gentleman. London, Printed
by R. and W. Leybourn, for Thomas Johnson at the Golden Key in St.
Pauls Church-yard. M.DC.LVII.
[39]Jer. Collier, The Great Historical, Geographical, Genealogical and
Poetical Dictionary, Vol. II, London, 1701, art. “Oughtred.”
[40]Rigaud op. cit., Vol. I, p. 82.
[41]A. De Morgan, Budget of Paradoxes, London, 1872, p. 451; 2d ed.,
Chicago, 1915, Vol. II, p. 303.
[42]E. Gunter, Description and Use of the Sector, the Crosse-staffe and
other Instruments, London, 1624, second book, p. 31.
[43]F. Cajori, “On the History of a Notation in Trigonometry,” Nature,
Vol. XCIV, 1915, pp. 642, 643.
[44]A. von Braunmühl, Geschichte der Trigonometrie, 2. Teil, Leipzig,
1903, pp. 42, 91.
[45]H. Hankel, Geschichte der Mathematik in Alterthum und Mittelalter,
Leipzig, 1874, pp. 369, 370.
[46]M. Cantor, Vorlesungen über Geschichte der Mathematik, II, 1900, pp.
640, 641.
[47]This matter has been discussed in a paper by F. Cajori, “A History of
the Arithmetical Methods of Approximation, etc., Colorado College
Publication, General Series No. 51, 1910, pp. 182-84. Later this
subject was again treated by G. Eneström in Bibliotheca mathematica,
3. Folge, Vol. XI, 1911, pp. 234, 235.
[48]See F. Cajori, op. cit., p. 193.
[49]See William Oughtred’s Key of the Mathematicks, London, 1694, pp.
173-75, tract, “Of the Resolution of the Affected Equations,” or any
edition of the Clavis after the first.
[50]A. De Morgan, op. cit., p. 451; 2d ed., Vol. II, p. 303.
[51]See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909,
pp. 7-14, Addenda, p. ii.
[52]Rigaud, op. cit., Vol. I, p. 12.
[53]The New Artificial Gauging Line or Rod: together with rules
concerning the use thereof: Invented and written by William Oughtred,
London, 1633.
[54]W. Oughtred, Apologeticall Epistle, p. 13.
[55]Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, (1915),
p. 169. In this article Glaisher republishes the “Appendix” in full.
[56]Aubrey, op. cit., Vol. II, 1898, p. 108.
[57]Wood’s Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247.
[58]Wood, op. cit., Vol. II, p. 445.
[59]Rigaud, op. cit., Vol. I, pp. 33, 35.
[60]Rigaud, op. cit., Vol. I, pp. 16, 26.
[61]Rigaud, op. cit., Vol. I, p. 66.
[62]Ibid., Vol. I, p. 9.
[63]Rigaud, op. cit., Vol. II, p. 475.
[64]Ibid., Vol. II, p. 471.
[65]J. W. L. Glaisher, “On Early Logarithmic Tables, and Their
Calculators,” Philosophical Magazine, 4th Ser., Vol. XLV (1873), pp.
378, 379.
[66]Rigaud, op. cit., Vol. I, p. 65.
[67]Rigaud, op. cit., Vol. I, p. 87.
[68]King’s Life of John Locke, Vol. I, London, 1830, p. 227.
[69]Exercitationum Mathematicarum Decas prima, Naples, 1627, and probably
Cataldus’ Transformatio Geometrica, Bonon., 1612.
[70]Rigaud, op. cit., Vol. II, pp. 477-80.
[71]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster,
Sometimes publike Professor of Astronomie in Gresham Colledge in
London, by John Twysden, London, 1659.
[72]The Works of the Honourable Robert Boyle in five volumes, to which is
prefixed the Life of the Author, Vol. I, London, 1744, p. 24.
[73]The letter is printed in John Wallis’ De algebra tractatus, 1693, p.
206.
[74]See La Correspondance de Descartes, published by Charles Adam and
Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457.
[75]H. Bosmans, S.J., “La première édition de la Clavis Mathematica
d’Oughtred. Son influence sur la Géométrie de Descartes,” Annales de
la société scientifique de Bruxelles, 35th year, 1910-11, Part II, pp.
24-78.
[76]Ibid., p. 78.
[77]Vincent Wing, Harmonicon coeleste, London, 1651, p. 5.
[78]Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenta
inquisitio brevis, Oxford, 1653, p. 7.
[79]John Wallis, Elenchus geometriae Hobbianae, Oxford, 1655, p. 48.
[80]An Idea of Arithmetick, at first designed for the use of the Free
Schoole at Thurlow in Suffolk. . . . . By R. B., Schoolmaster there,
London, 1655, p. 6.
[81]The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster
. . . . by John Twysden, London, 1659, p. 1.
[82]Moor’s Arithmetick in two Books, London, 1660, p. 89.
[83]Isaac Barrow, Euclidis data, Cambridge, 1657, p. 2.
[84]Francisci Dulaurens Specima mathematica, Paris, 1667, p. 1.
[85]Elémens des mathématiques, Paris, 1675, Preface signed “J. P.”
[86]Nouveaux élémens de géométrie, Paris, 1692 (permission to print
1684).
[87]Ozanam, Dictionnaire mathématique, Paris, 1691, p. 12.
[88]Analyse des infiniment petits, Paris, 1696, p. 11.
[89]Petro Nicolas, De conchoidibus et cissoidibus exercitationes
geometricae, Toulouse, 1697, p. 17.
[90]R. P. Bernard Lamy, Elémens des mathématiques, Amsterdam, 1692
(permission to print 1680).
[91]Nouveaux élémens de géométrie, 2d ed., The Hague, 1690, p. 304.
[92]W. W. Beman in L’intermédiaire des mathématiciens, Paris, Vol. IX,
1902, p. 229, question 2424.
[93]John Collins, The Mariner’s Plain Scale New Plain’d, London, 1659, p.
25.
[94]James Gregory, Optica promota, London, 1663, pp. 19, 48.
[95]Philosophical Transactions, Vol. III, London, p. 868.
[96]William Leybourn, The Line of Proportion, London, 1673, p. 14.
[97]Elementa geometriae . . . . a Gulielmo Sanders, Glasgow, 1686, p. 3.
[98]Cocker’s Decimal Arithmetick, . . . . perused by John Hawkins,
London, 1695 (preface dated 1684), p. 41.
[99]Joseph Raphson, Analysis Aequationum universalis, London, 1697, p.
26.
[100]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford,
1698, p. 107.
[101]John Ward, A Compendium of Algebra, 2d ed., London, 1698, p. 62.
[102]Plain Elements of Geometry and Plain Trigonometry, London, 1701, p.
63.
[103]George Shelley, Wingate’s Arithmetick, London, 1704, p. 343.
[104]A Synopsis of Algebra, Being a posthumous work of John Alexander of
Bern, Swisserland. . . . . Done from the Latin by Sam. Cobb, London,
1709, p. 16.
[105]John Craig, De Calculo fluentium, London, 1718, p. 35. The notation
A:B::C:D is given also.
[106]Trigonometry, 2d ed., Edinburgh, 1724, p. 11.
[107]Méthode pour la mésure des surfaces, la dimension des solides . . .
. par M. Carré de l’académie r. des sciences, 1700, p. 59.
[108]Application de l’algèbre à géométrie . . . . Paris, 1705.
[109]Elémens de la géométrie de l’infini, by M. de Fontenelle, Paris,
1727, p. 110.
[110]Eclaircissemens sur l’analyse des infiniment petits, by M. Varignon,
Paris, 1725, p. 87.
[111]Application de la géométrie ordinaire et des calculs différentiel et
intégral, by M. Robillard, Paris, 1753.
[112]Traité de géométrie théorique et pratique, new ed., Paris, 1764, p.
15.
[113]Recherches sur les courbes à double courbure, Paris, 1731, p. 13.
[114]Analyse des infiniment petits, by the Marquis de L’Hospital. New ed.
by M. Le Fèvre, Paris, 1781, p. 41. In this volume passages in fine
print, probably supplied by the editor, contain the notation a:b::c:d;
the parts in large type give Oughtred’s original notation.
[115]The tendency during the eighteenth century is shown in part by the
following data: Jacobi Bernoulli Opera, Tomus primus, Geneva, 1744,
gives B.A::D.C on p. 368, the paper having been first published in
1688; on p. 419 is given GE:AG=LA:ML, the paper having been first
published in 1689. Bernhardi Nieuwentiit, Considerationes circa
analyseos ad quantitates infinitè parvas applicatae principia,
Amsterdam, 1694, p. 20, and Analysis infinitorum, Amsterdam, 1695, on
p. 276, have x:c::s:r. Paul Halcken’s Deliciae mathematicae, Hamburg,
1719, gives a:b::c:d. Johannis Baptistae Caraccioli, Geometria
algebraica universa, Rome, 1759, p. 79, has a.b::c.d. Delle corde
ouverto fibre elastiche schediasmi fisico-matematici del conte
Giordano Riccati, Bologna, 1767, p. 65, gives P:b::r:ds. “Produzioni
mathematiche” del Conte Giulio Carlo de Fagnano, Vol. I, Pesario,
1750, p. 193, has a.b::c.d. L. Mascheroni, Géométrie du compas,
translated by A. M. Carette, Paris, 1798, p. 188, gives
√(3):2::√(2):Lp. Danielis Melandri and Paulli Frisi, De theoria lunae
commentarii, Parma, 1769, p. 13, has a:b::c:d. Vicentio Riccato and
Hieronymo Saladino, Institutiones analyticae, Vol. I, Bologna, 1765,
p. 47, gives x:a::m:n+m. R. G. Boscovich, Opera pertinentia ad opticam
et astronomiam, Bassani, 1785, p. 409, uses a:b::c:d. Jacob Bernoulli,
Ars Conjectandi, Basel, 1713, has n-r.n-1::c.d. Pavlini Chelvicii,
Institutiones analyticae, editio post tertiam Romanam prima in
Germania, Vienna, 1761, p. 2, a.b::c.d. Christiani Wolfii, Elementa
matheseos universae, Vol. III, Geneva, 1735, p. 63, has AB:AE=1:q.
Johann Bernoulli, Opera omnia, Vol. I, Lausanne and Geneva, 1742, p.
43, has a:b=c:d. D. C. Walmesley, Analyse des mesures des rapports et
des angles, Paris, 1749, uses extensively a.b::c.d, later a:b::c:d. G.
W. Krafft, Institutiones geometriae sublimoris, Tübingen, 1753, p.
194, has a:b=c:d. J. H. Lambert, Photometria, 1760, p. 104, has C:π
=BC²:MH². Meccanica sublime del Dott. Domenico Bartaloni, Naples,
1765, has a:b::c:d. Occasionally ratio is not designated by a.b, nor
by a:b, but by a, b, as for instance in A. de Moivre’s Doctrine of
Chance, London, 1756, p. 34, where he writes a, b::1, q. A further
variation in the designation of ratio is found in James Atkinson’s
Epitome of the Art of Navigation, London, 1718, p. 24, namely,
3..2::72..48. Curious notations are given in Rich. Balam’s Algebra,
London, 1653.
[116]Chr. Clavii Operum mathematicorum tomus secundus, Mayence, 1611,
Algebra, p. 39.
[117]Invention nouvelle en l’algèbre, by Albert Girard, Amsterdam, 1629,
p. 17.
[118]La géométrie et pratique générale d’icelle, par I. Errard de
Bar-le-Duc, Ingénieur ordinaire de sa Majesté, 3d ed., revised by D.
H. P. E. M., Paris, 1619, p. 216.
[119]Novae geometriae clavis algebra, authore P. Jacobo de Billy, Paris,
1643, p. 157; also an Abridgement of the Precepts of Algebra. Written
in French by James de Billy, London, 1659, p. 346.
[120]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster,
Sometime publike Professor of Astronomie in Gresham Colledge in
London, London, 1659, p. 7.
[121]Quarterly Jour. of Pure and Applied Math., Vol. XLVI (London, 1915),
p. 191.
[122]Pietro Cossali, Origine, trasporto in Italia primi progressi in essa
dell’ algebra, Vol. I, Parmense, 1797, p. 52.
[123]In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio
brevis, Auctore Setho Wardo, Oxford, 1653, p. 1.
[124]John Wallis, Algebra, London, 1685, p. 321, and in some of his other
works. He makes greater use of Harriot’s symbols.
[125]Euclidis data, 1657, p. 1; also Euclidis elementorum libris XV,
London, 1659, p. 1.
[126]John Kersey, Algebra, London, 1673, p. 321.
[127]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford,
1698, p. 142.
[128]Cocker’s Decimal Arithmetick, perused by John Hawkins, London, 1695
(preface dated 1684), p. 278.
[129]Th. Baker, The Geometrical Key, London, 1684, p. 15.
[130]Richard Sault, A New Treatise of Algebra, London (no date).
[131]Richard Rawlinson in a pamphlet without date, issued sometime
between 1655 and 1668, containing trigonometric formulas. There is a
copy in the British Museum.
[132]F. Dulaurens, Specima mathematica, Paris, 1667, p. 1.
[133]J. Milnes, Sectionum conicarum elementa, Oxford, 1702, p. 42.
[134]Cheyne, Philosophical Principles of Natural Religion, London, 1705,
p. 55.
[135]J. Craig, De calculo fluentium, London, 1718, p. 86.
[136]Jo. Wilson, Trigonometry, 2d ed., Edinburgh, 1724, p. v.
[137]Commercium Epistolicum, 1712, p. 20.
[138]C. Le Paige, “Sur l’origine de certains signes d’opération,” Annales
de la société scientifique de Bruxelles, 16th year, 1891-92, Part II,
pp. 79-82.
[139]Gravelaar, “Over den oorsprong van ons maalteeken (×),” Wiskundig
Tijdschrift, 6th year. We have not had access to this article.
[140]H. Bosmans, op. cit., p. 40.
[141]Claudii Ptolemaei . . . . annotationes, Bâle, 1551. This reference
is taken from the Encyclopédie des sciences mathématiques, Tome I,
Vol. I, Fasc. 1, p. 40.
[142]Due Correction for Mr. Hobbes. Or Schoole Discipline, for not saying
his Lessons right. In answer to his Six Lessons, directed to the
Professors of Mathematicks. By the Professor of Geometry. Oxford,
1656, pp. 7, 47, 50.
[143]Oughtred, Apologeticall Epistle, p. 27.
[144]J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes,
London, 1850, pp. 279-92.
INDEX
Adam, Charles, 71
Agnesi, Maria G., 77
Alexander, J., 76
Allen, E., 35
Analysis, 19, 20
Apollonius of Perga, 20, 85
Archimedes, 18, 20, 85
Aristotle, 69
Ashmole, E., 13
Atkinson, J., 79
Atwood, 56
Aubrey, 3, 7, 8, 12-16, 58, 59
Austin, 58
Baker, T., 82
Balam, R., 79
Bar-le-Duc, de, 80
Barrow, S., 1, 32, 73, 74, 80, 81
Bartaloni, D., 79
Beman, W. W., 74, 75
Bernoulli, Jakob, 78-80
Bernoulli, John, 79, 80
Billingsley’s Euclid, 15
Billion, 20
Billy, Jacobo de, 80
Binomial formula, 25, 29
Bliss, P., 60
Boscovich, R. G., 78
Bosmans, H., 72, 83
Boyle, R., 1, 69
Braunmühl, von, 39
Brearly, W., 59
Briggs, 6, 36, 55
Brookes, Christopher, 7, 53, 59
Cajori, F., 27, 39, 40, 47
Cantor, M., 40, 41
Caraccioli, J. B., 78
Cardan, 71
Carré, 77
Carrete, N. M., 78
Caryll, C., 7
Cataldi, 67
Cavalieri, 65, 66
Cavendish, Charles, 17, 62, 66
Charles I, 9, 60
Chelvicius, P., 79
Cheyne, G., 82
_Circles of Proportion_, 35, 37, 48, 49, 51, 59, 87, 88
Clairaut, 77
Clark, A., 3
Clark, G., 63
Clarke, F. L., 3
_Clavis mathematicae_, 1, 5, 10, 14, 17-35, 45, 46, 51, 57-63,
68-73, 81, 85, 87
Clavius, 26, 80
Clerc, le, 77
Cobb, S., 76
Cocker, 76, 82
Collins, John, 15, 19, 63, 64, 67, 68, 76, 82
Colson, J., 77
Conchoid, 12
Conic sections, 11, 53
Cossali, P., 81
Cotes, R., 1, 85
Craig, J., 76, 82
Cross, symbol of multiplication, 27, 38, 55, 56, 82, 83
Cubic equations, 28, 34, 42, 45
Decimal fractions, notation of, 21
Degree, centesimal division, 39
Delamain, R., 4, 9, 10, 11, 47, 48, 51, 60, 84, 88, 89, 91, 93, 94
De Moivre, 32, 79
De Morgan, A., 5, 16, 37, 46, 47, 54
Descartes, R., 1, 25, 47, 57, 68-72, 80
Dibuadius, 79
Difference, symbol for, 27, 81
Diophantus, 63, 85
Division, abbreviated, 21, 23, 24
Dulaurens, F., 74, 82
Earl of Arundel, 10, 13, 15, 17
Edleston, J., 95
Eneström, G., 40
Equations, solution of, 18, 28, 29, 31, 34, 39-45, 87
Errard de Bar-le-Duc, 80
Eton College, 3, 4
Euclid, 1, 15, 18, 20, 25, 27, 28, 79, 81, 83, 85
Euler, L., 37, 39
Ewart, 59
Exponents, 25, 28, 29
Fagnano, de, 78
Flower, 56
Fontenelle, de, 77
Forster, W., 35, 48, 59, 88
Foster, S., 27, 67, 69, 73, 80, 89
Frisi, P., 78
Gascoigne, 59, 61
Gauss, C. F., 48
Geysius, 67
Ghetaldi, 70, 71
Gibson, 67
Girard, A., 32, 80
Glaisher, J. W. L., 54-56, 64, 80
Glorioso, 67, 68
_Grammelogia_, 4, 47, 89
Gravelaar, 83
Greater than, symbol for, 81
Greatrex, R., 15
Gregory, D., 32
Gregory, J., 27, 76
Gresham College, 1, 6, 27, 59, 61, 80
Guisnée, 77
Gunter, E., 37, 47, 86
Gunter’s scale, 37
Halcken, P., 78
Hales, J., 7
Halley, E., 1, 18, 69
Hankel, H., 40
Harper, T., 18
Harriot, T., 45, 47, 57, 58, 69-71, 81
Harris, J., 76
Hartlib, 69
Haughton, A., 35, 59
Hawkins, J., 76, 82
Hearn, 56
Helmholtz, 48
Henry, J., 48
Henry van Etten, 52, 53
Henshaw, T., 8, 58, 61
Hobbes, 73, 86
Hollar, 14
Holsatus, 13
Hooganhuysen, 64
Hooke, Rb., 1
Horner’s method, 45
Horology, 18, 50
Horrox, J., 4
Hospital, de l’, 74, 77
Howard, Th. _See_ Earl of Arundel.
Howard, W., 17, 18, 59
Hutchinson, A., 6
Invisible college, 1
Joule, 48
Kepler, J., 6
Kersey, J., 32, 73, 82
Keylway, R., 65
King, 67
Kings College, Cambridge, 3, 35
Krafft, G. W., 79
Lambert, J. H., 79
Lamy, R. P. B., 74
Laud, Archbishop, 65
Leake, W., 53
Le Clerc, 77
Leech, W., 59
Le Fèvre, 77
Leibniz, 47, 78, 80
Leonelli, 56
Le Paige, de, 83
Less than, symbol for, 81
Leurechon, 52
Leybourn, 35, 64, 76
Lichfield, Mrs., 19
Lilly, W., 8, 9
Locke, J., 67
Logarithms, 6, 21, 27, 28, 38, 39, 42, 46, 54-56, 65, 92, 93;
natural, 55;
radix method of computing, 55, 56
Lower, W., 58
Ludolph à Ceulen, 79
Manning, 56
Manning, O., 7, 8, 13-15
Mascheroni, L., 78
Mayer, R., 47
Melandri, D., 78
Mercator, N., 13
Mersenne, 63
Milbourn, W., 45
Million, 20
Milnes, J., 82
Moivre, de, 32, 79
Moore, Jonas, 32, 54, 58, 73
Moreland, S., 70
Morse, R., 48
Multiplication, abbreviated, 21, 22, 24;
symbol for, 27, 82, 83
Mydorge, 54
Napier, J., 6, 7, 21, 27, 38, 39, 52, 54, 57, 59
Napier’s analogies, 39
Newton, Sir Isaac, 1, 25, 29, 40, 41, 45, 47, 59, 65, 86, 92-95
Nichols, J., 6, 14
Nicolas, R. P. P., 74
Nieuwentiit, B., 78
Norwood, R., 37, 38, 80
_Opuscula mathematica hactenus inedita_, 16, 21, 75
Orchard, 56
_Oughtredus explicatus_, 64
Ozanam, 74
π, symbol for, 32
Paige, C. de, 83
Pardies, 76
Parentheses, 26, 79, 80
Partridge, S., 47
Peano, 86
Perfect number, 41
Pitiscus, 15
Planisphere, 53, 92, 93
Prestet, J., 74
Price, 11
Proportion, notation for, 26, 27, 73-79
Protheroe, 58
Ptolemy, 83
Quadratic equation, 29, 31, 34
Radix method, 55, 56
Rahn, 27
Raphson, J., 40, 41, 76
Ratio, notation of, 21, 73-80
Rawlinson, R., 39, 82
Regula falsa, 18
Regular solids, 18
Riccati, G., 78
Riccati, V., 78
Rigaud, 7, 12, 13, 19, 48, 61-66, 68
Robillard, 77
Robinson, W., 13, 48, 59, 62, 63
Rooke, L., 59, 61
Saladini, H., 78
Sanders, W., 76
Sault, R., 82
Scarborough, Charles, 16, 54, 58, 60
Schooten, Van, 1
Schreshensuchs, O., 83
Scratch method, 23
Shakespeare, 52
Shelley, G., 76
Shipley, A. E., 1
Shuttleworth, 59
Slide rule, 9, 46-49, 50, 60, 88, 93
Smethwyck, 58
Smith, J., 50
Snellius, W., 79
Solids, regular, 18
Speidell, John, 38, 55
Spherical triangles, 53, 54, 93
Stokes, R., 35, 36, 58
Sudell, 59
Sun dials, 5, 9, 50, 51, 52, 60, 92
Tannery, P., 71
Todhunter, 60
Torporley, 58
Triangles, spherical, 53, 54, 93
_Trigonometria_, 21, 36, 55, 75
Trigonometric functions, symbols for, 36, 37, 55, 56
_Trigonometrie_, 21, 35, 39
Trisection of angles, 28
Twysden, 59, 68, 69, 73
Varignon, 77
Vieta, 1, 2, 25, 32, 33, 35, 39-41, 45, 63, 67, 70, 71
Vlack, 65
Von Braunmühl, 39
Wadham College, 5, 53
Wallis, John, 1, 19, 27, 33, 45, 57-59, 63, 64, 66-74, 79-81, 86
Walmesley, D. C., 79
Ward, Bishop, 13
Ward, John, 76
Ward, Seth, 55, 58, 60, 68, 73, 74, 81
Watch-making, 18, 50
Weber, W. E., 48
Weddle, 56
Wells, E., 76, 82
Wharton, 60
Whitlock, B., 8, 9
Wilson, J., 77, 82
Wing, V., 73, 75
Wingate, E., 32, 47, 73
Wolf, Christian, 79
Wood, A., 60, 61
Wood, R., 18, 59
Wren, Christopher, 5, 58, 59, 76
Wright, E., 6, 27, 38, 54
Wright, S., 54
Transcriber’s Notes
A handful of typos, mostly misplaced punctuation, were silently
corrected.
HTML and UTF text versions make heavy use of mathematical symbols:
particularly superscripts, subscripts, and combining characters. Some
viewers may require user assistance to find fonts containing these
characters.
The text versions miss much of the formatting, especially in mathematical
formulas:
--Several arithmetic examples must be viewed in a monospaced font (which
recognizes combining characters) to be legible.
--Formulas under the horizontal line of a square root symbol are
parenthesized.
--Subscripts are preceded by “_”.
--Superscripts are preceded by “^”.
--Italics, used primarily in formulas and bibliographical entries, are
not indicated in the text.
--Italics in the index are delimited by “_”.
--Underlines are not indicated (in particular, in the fractional part of
a decimal number in Oughtred's notation).
--The idiosyncratic “greater than” and “less than” symbols are indicated
as {symbol} in the ASCII version.
--Overdots, underdots, and slashmarks around digits (in the long division
example) are not indicated in the ASCII version.
--Greek letters are spelled out within {curly brackets} in the ASCII
version.