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William Shepherd.

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the " Elements" only, able to give a satisfactory account of
every step in the demonstrations which they have been calle'd
on to perform. Tliis we think it right to state, that unwarrant-
able prejudices may not be excited against a book that has
been in deservedly high reputation so many centuries.

Mr. Reynard, in describing his labours, says, that in order to
throw some light on the contents, and to give the student an
idea of what he is going to do, and what advantages he will
derive, an introduction immediately precedes each book, and
by way of exercise, and to try the test of the student's know-
ledge, after he has read through each book (of which there
are^^ght), thei^ is a suitable collection of questions, which
will give the mind ample scope for its exercise, and will af-
ford the best criterion of his industry and attention. Mr. R.
states likewise, that he has introduced, in the progress of his
work, some new theorems, and that all the theorems through
the work are demonstrated by the direct method, u-hich he
imagines is the safest way of proceeding. If it be the safest
method, we cannot help observing, that the reductio ad ab-
surdum very frequently carries with it a conviction equally
satisfactory to the mind with the direct method of proof.
Our author has added at the end, a selection of miscellaneous
questions, chiefly taken from different authors, some of
which will demand the ingenuity and knowledge of the skilfiil



470 MATHEMATICS.

geometrician ; many are likewise given, with an intention
that he may, at the same time, be exercised in solving them
analytically. These supplementary questions, and the Ques-
tioues Solvendae at the end of each book, are applicable to
almost any other work of Geometry, and may be found useful
to those who study this branch of science by means of Sim-
son's, or any other Euclid, or by Simpson's, Bonnycastle's,
or the other introductions to Geometry.

" The Elements of Plane Geometry ; containing the first
six books of Euclid, from the text of Dr. Simson, with notes
critical and explanatory ; to which are added, book vii. in-
cluding several important propositions which are not in Eu-
clid ; book viii. consisting of Practical Geometry ; also book
ix*. of Planes and their intersections ; and book x. of the
Geometry of Solids." By Thomas Keith. 1814.

This work, which has not been long before the public, ap-
pears to be a very excellent introduction to mathematical
knowledge. The notes to the first six books are brief, but
much to the purpose, and such as will assist the student in
surmounting the little difficulties that may present themselves
to him. They are intended to elucidate, improve, or extend
the text. The seventh, ihe author tells us, is an expanded
epitome of the theorems in the first six books of Euclid, ar-
ranged in the order which the nature of the subject appeal^ to
require : it contains some propositions from the tenih, twelfth,
and thirteenth books of Euclid, besides a number that are
not in his work, some of which are from Pappus, of Alexan-
dria. To these are added, a few propositions relative to the
rectification and quadrature of the circle. " Throughout the
whole perfonnance," says Mr. Keith, " the utmost care has
been taken to render the several subjects accurate, plain,' and
intelligible to young students. If the author has succeeded
in these particulars, his purpose will be answered : and tliere-
fore he submits, without apprehension, to scientifical men,
the result of his endeavours to elucidate the doctrine and
extend the iisefubess of Euclid's Elements." We give him



GEOMETRY. 471

full credit for his intentions, and thinL it will be admitted that
he has succeeded.

Thus have we taken a review of a great variety of elemen-
tary works on the science of geometry, of which any one will
be sufficient as an introduction to that highly important branch
of knowledge. As, however, we mtend our work for real
practical use, we shall offer a few directions respecting the
best method of studying geometry. These directions will
apply to any of the works which have been mentioned.

In the study of history, and indeed of almost every branch
of the belles lettres, the student derives considerable pleasure
from the pursuit, even in its first stages. The knowledge of
new facts, the conterrtplation of the diversified customs and
manners of different people, or of the same people in the va-
rious stages of civilization, afford at once, adiusement com-
bined with uistruction.

The case is different in mathematics : many a youth has
taken Euclid into his hands, or perhaps some easier treatise
on the Elements of Geometry ; and has read a book or two,
without perceiving even tlie drift of his author. That such
should throw away their books in disgust can be no matter of
surprize ; though it is probable that with proper directions
' they might have become good geometricians. To persons of
this description, who can derive no benefit from instructors, and
who have no scientific friend at hand whom they can consult,
our work may stand in the stead of a living monitor.

In the study of geometry, care must first be taken to com-
mit perfectly to memory, the definitions, axioms, and postu-
lates, because to these, reference is perpetually made.

The propositions are to be carefully read and thoroughly
understood. The learner must not advance to the second till
he is sure that he is master of the first, and so of the rest
After he has thus gone through the first book, it will be an
useful and highly advantageous exercise, if he return and com-
mit the enunciation of each proposition to his memory, and
then, either having the figure before him, or drawing it out



478 MATHEMATICS. *

for himself, write out a demonstration in his own word*.
This he cannot do without being sure that he understands
what he is about ; and till he can do this, he ought not to
leave a single proposition. This method, we will venture to
affirm, is, to persons of common capacities, the best and
surest to attain a well-grounded knowledge in geometry. It .
is that mode recommended by the Latins, " Festina lente."
Slow at first, but sure. And after 'the two first books are
thus gone through, the remainder will present to the youth
great pleasure, and but fev/ difficulties.

The doctrine of ratios in the fifth book, may perhaps
retard the learner awhile : it is treated of in Simson's Euclid,
in a method less easy than in the several treatises of the
Elements of Geometry which we have enumerated, to any
one of which the reader of Euclid may be referred for assist-
ance. In the solid geometry, if he would call in the aid of
the senses, which is not absolutely jiecessary, he wjll do well ,
to have Cowley's figures before him.

The learner will scarcely enter into the spirit of the geo-
metrical notes till he is master of the elements. Euchd's
data in the edition of Dr. Simson, Barrow, &c., the maxima
and minima, in that of Mr. Thomas Simpson, and the other
appendages to the different treatises in this science, will afford
the young geometrician much intellectual pleasure, with the
expense of little labour, if he has laid a solid foundation in
the way recommended.

The pupil having made himself master of the Elementary
parts of Algebra and Geometry, will be fully prepared to
apply his knowledge in one branch of science to the solution
of problems in the other. The application of algebra to geo-
metry is of two kinds : that which regards the plane or com-
mon geometry, and that which respects what is usually deno-
minated the higher geometry, or the nature of curve lines.
The first of these is concerned in the algebraical solutions of
geometrical problems, and in the investigation of theorems in
gcomctripal figures, by means of algebraical investigations qv



GEOMETRY. 473

demonstrations. This method of resolving geometrical pro-
blems is, in many cases, more direct and easy, than that of
geometrical analysis ; but of course the solution in this way,
depends upon a previous acquaintance with the method of
expressing geometrical magnitudes, as well as their mutual
positions and relations, by algebraical notation, that is^ a \\pe
is represented by a single letter, a rectangle by the product of
two letters, expressing its sides, a rectangular parallepipedon by
the product of three letters; two of which will represent the su-
perficies of the base, and the other its perpendicular altitude :
thus, if a and b represent the sides of a rectangle, then ax b
or a b will represent the superficies ; and if a prison be erected
on that superficies, whose altitude is denoted by c, then the
solid contents will be expressed hy axbxc or a be. The
opposite position of straight lines may be expressed by the
signs + and , and segments of lines may be denoted by
letters with these signs prefixed, as circumstances require.

In the solution of problems, the following general observa-
tions should be attended to. When any geometrical problem is
proposed for algebraical resolution, we must in the first place,
describe a figure that shall represent the parts or conditions
of the problem ; then, having considered the nature of the
problem, the figure must be prepared for solution, if neces-
sary, by producing and drawing such lines as appear most
conducive to that purpose. Then the lines known and un-
known are to be represented by proper symbols, and the
operation to be proceeded on, by. observing the relation which
4he several parts have to each other. No general rules can
be laid down for drawing the lines, and selecting the most
proper quantities to substitute for them, so as always to bring
out the most simple and direct conclusions. The following
general directions, taken chiefly from the Algebra and Select
Exercises of Mr. Thomas Simpson, are given and referred to
by almost all English writers on this subject.

1. In preparing the figure, by drawing lines, let thena be
either parallel or perpendicular to other lines in the figure, or



474 MATHEMATICS.

so as to form similar triangles ; and if an angle be given, let
the perpendicular be opposite to that angle, and fall from the
end of the given line, if possible.

2. In selecting proper quantities for substitution, let those
be chosen, whether required or not, which lie nearest the
kno%Mi or given parts of the figure/ and by means of which
the next adjacent parts may be expressed, without the inter-
vention of surds, by addition and subtraction only. Thus, if
the problem were to find the perpendicular of a plane triangle,
from the three sides given, it will be much better to substitute
for one of the segments of the base, than for the perpendicu-
lar, though that be the quantity required ; because the whole
base being given, the other segment will be given, or ex-
pressed, by subtraction only, and so the final equation will
come out a simple one ; from whence the segments being
known, the perpendicular is easily found by common arithme-
tic ; whereas, if that perpendicular were first sought, both the
segments would be surd quantities, and the final equation a
quadratic one.

3. Where, in any problem, there are two lines or quantities
alike related to other parts of the figure or problem, the best
way is to make use of neither of them, but to substitute for
their sum, their rectangle, or th(5 sum of their alternate quo-
tients, or for some line or lioetf in the figure, to which they
both have the same relation. J

4. If the area, or the perimeter of a figure, be given, or
such parts of it as have but a remote relation to the parts re-
quired, it will be sometimes of use to assume another figure
similar to the proposed one, of which one side is unity, or
some other known quantity, whence the other parts of the
figure may be found, and an equation obtained.

The writers on this branch of science are numerous, to
several of which we have already referred. In Simpson's
Algebra, and the second part of his Select Exercises, there
are a good number of Problems. A few pages of Mr.
Frend's Algebra are devoted to it, but these are scarcely



GEOMETRY. 475

sufficient to give the pupil an insight into the subject. In
the second volume of Mr. Bonnycastle's larger v^'ork on
Algebra, will be found a variety of problems relating to
plane geometry, and also to tlie doctrine of curves. Al-
most all the volumes of the " Ladies' Diary," and the " Ma-
thematical Repository," by Mr. Leybourn, will furnish the
Mathematical student with abundance of work, from the
easiest to the abstrusest problems. The most interesting
parts of the " Ladies* Diary," from 1704 to 1773, were select-
ed and published in five volumes, by Dr. Huttou, in 1775.
To which the learned editor added a sixth volume, entitled
" Miscellanea Mathematica," consisting of a large collection
of curious mathematical problems, and their solutions, &c.
Mr. Leybourn 's Mathematical Repository consists of three
volumes, 12mo.

The mathematical student will find an advantage in possess-
ing Hutton's, or Barlow's Dictionary, or Nicholson's " Brit-
ish Encyclopedia ;" or, above all, the great national work of
the Rev. Dr. Rees, entitled, *' The New Cyclopedia," of
which nearly sixty parts are before the public.



CHAP. XXXI.



MATHEMATICS,

Continued,



Trigonometry History of this branch of Science Ancient writers on
Trigonometrj' : Theodosius Menelaus Ptolemy Purback Regio-
moutanos Copernicus MaaroHcus Rheticus O tho Pitiscus Clu-
vius Ceulen Napier Briggs Guntcr Trigonometry, on whatfound-
cd Methods of investigating its principles Practical rulps. Modern
authors: Martin Ashworth Sim&on Simpson Vince Leslie Wood-.
house Bonnycastle Keith Kelly ; and Walker.

1 HE art of measuring tlie sides and angles of a triangle,
whether plane or spherical, is called plane or spherical Trigo-
nometry, This is an art of the greatest use in the mathema-
tical sciences, especially in navigation, surveying, levelling,
diahng, and geography. By means of trigonometry, we come
to know the magnitude of the earth, the moon, the planets, and
even the sun : their distances from us, and from one another ;
their motions and several occultations and eclipses. Hence it
is probable, that this art was cultivated from the earliest ages
of mathematical knowledge, though no records have been left
us, by which we can trace it to a higher age than that of
Hippardms, who flourished about a century and a half before
the commencement of the Christian aera ; and who is reported



TRIGONOMETRY. 477

by Theon, in his Commentary on Ptolemy's Almagest, to
have written a work, in twelve books, on the chords of circu-
lar arcs, which was no doubt a treatise on Trigonometry.
The earliest existing work on the subject, is the spherics of
Theodosius, a native of Tripoli, in Bithynia, who, in the
times of Cicero and Pompey, collected the scattered principles
of science, which had been discovered by his predecessors,
and formed them into a regular treatise, in three books, con-
taining a variety of the most necessary and useful propositions
relating to the sphere, arranged and demonstrated with great
perspicuity and elegance, after the manner of Euclid's Ele-
ments. The first of these books contained twenty-two propo-
sitions ; the second, twentyrthree ; and the third, fourteen.
They were translated by the Arabians out of the original
Greek, into their own language : from which, the work was
translated into Latin, and printed at Venice. The Arabic
version was found afterwards to be very defective, and a more
complete edition >fas published in Greek and Latin at Paris,
in 1558. De Chales printed it in his Cursus Mathematicus.
The edition most generally referred to at this time, is that
by Dr. Barrow, published in 1675, illustrated and demon-
strated in a new and corfcise method. There is a good Oxford
edition, in 8vo, 1709.

The next of the Greek writers, after Theodosius, who has
professedly treated on this subject, is Menelaus, an astrono-
mer and mathematician of considerable eminence, who flou-
rished during the first century of the Christian aera. He was
author of three books " On Spherics," which have come
down to us through the medium of the Arabic language. A
Latin version of this work was published at Paris, by Father
Mersenne, in 1664, with corrections, restorations, and addi-
tional propositions. The Spherics of Menelaus contain besides
the first principles of the science, a number of propositions of
a more difficult kind. Dr. Halley prepared a new' edition of
this work, which was published in Svo, J 758, by Costard,
the author of the " History of Astronomy." Menelaus is



478 MATHEMATICS.

said to have been the author of another work on the subtenses,
or chords, of circular arcs, being, probably, a treatise on the
ancient method of constructing trigonometrical tables, which
has not come do\vn to the modems. This loss has been re-
paired in a good measure by Ptolemy, who, in the first book
of his Almagest, published early in the second century after
Christ, has given a table of arcs and their chords, to every
half degree of the semicircle, in the formation of which he
divides the radius and the arc whose chord is equal to radius,
each into sixty equal parts; and then estimates all other arcs
by sixtieths of that arc, and the chords by sixtieths of that
chord, or of tlie radius. Ptolemy is said to have been the
autlior of the proposition " that the rectangle of the two
diagonals of any quadrilateral inscribed in a circle, is equal to
the sum of the rectangles of its opposite sides."

After Ptolemy, and his commentator Theon, Jittle more is
known on this subject, till about the close of the eighth cen-
tury after Christ, when the ancient method of computing by
the chords was changed for that of sines, which was first in-
troduced into the science by the Arabians, who furnished the
several axioms and theorems, which are at present considered
as the foundation of trigonometry, as it is now taught. The
Arabians, though acquainted with the decimal scale of arith-
metic, did not in their trigonometry deviate from the Greeks
in the sexagesimal division of tlie radius, which continued in
use till the middle of the fifteenth century, when Purback, a
German, constructed a table of sines to a division of the
radius, into 600,000 equal parts, and computed them for
every ten minutes, in parts of this radius, by decimal notation.
The project was carried on by Regiomontanus, his pupil
and friend, who computed a table of sines, for every minute
of the quadrant to the radius, supposed to be divided into
100,000 parts.

Shortly after this, several other mathematicians contributed
to the advancement of this science, among whom was Wer-
ner, of Nuremburg, and Copernicus, the illustrious restorer



TRIGONOMETRY. 479

of the true system of the world, who wrote a brief treatise on
plane and spherical trigonometry, with a description and con*
struction of the canon of chords, which tract, together with a
table of sines, and their differences, for every ten minutes of
the quadrant, is inserted in his Revolutiones orbium ccclestium
published in 1543. Ten years after this, Erasmus Rheinold
published his Table of Tangents, and about the same time,
Maurolicus published a Table of Secants. The last named
author was one of the most considerable mathematicians of
his time. He was regarded by his learned contemporaries as
a second Archimedes, and was author of many very admirable
works.

The most complete work on the subject which had then
appeared, was a treatise, in two parts, by Vieta, printed at
Paris, in 1579 ; i" the first part of which he has given a table
of sines, tangents, and secants, for every minute of the quad-
rant, to radius of 100,000 with their differences ; and towards
the end of the quadrant, the tangents and secants are extended
to eight or nine places of figures. They are arranged like the
present tables. The second part of the volume, contains,
besides a regular account of the construction of the tables, a
compendious treatise on plane and spherical trigonometry,
with their application to a variety of curious subjects in geo-
metry, mensuration, &c., as likewise a number of particulars
relating to the quadrature of the circle, the duplication of the
cube and such like problems. This work of Vieta is said to
be extremely rare. It may be observed, though not strictly
connected with trigonometry, that Vieta gave in this work
computations of the ratio of the diameter of 'a circle to the
circumference, and of the length of the sine of one minute, to
several places of figures, by which he found that the sine of
one minute, is somewhere

Between 2908881959
And 2908882056;

and that the diameter of a circle being 1000, &c., the peri-



480 MATHEMATICS.

meter of the inscribed and circumscribed polygon of SQSjSlO
sides will be as follows : the

Perimeter of the inscribed polygon 31415926535
circumscribed 3 14 1 592fj537,

and of course that the circumference of the circle lies between
those two numbers. See Mensuration.

George Joachim Rheticus, a pupil of Copernicus, is the next
writer to be noticed on this subject. He formed the design of
computing the trigonometrical canon for every ten seconds of
the quadrant, to fifteeH places of decimals : he was able, how-
ever, to accomplish only that part of it relating to the sines
and co-sines. The work was finished by Valentine Otho, his
disciple, and published under the title of " Opus Palatinum
de Triangulis, in 1596." In this work, we have for the first
time, an entire table of sines, tangents, and secants, for every
ten seconds of the quadrant, to ten places of decimals, with their
differences. These tables were afterwards corrected by Bar-
tholomew Pitiscus, and he published his edition under the
title, of " Thesaurus Mathematicus," in l6l3. Previously to
this, Philip Lansberg published his " Geometria Triangu-
lorum," in four books, with the usual tables, and Pitiscus had
likewise given to the world, in 1599, his own work on Trigo-
nometry, which was reckoned a very complete work, both
with respect to the correctness of the tables, and its numerous
practical applications.

^'Christopher Cluvius, in the first volume of his works printed
at Mentz, 16 12, in five volumes folio, has given an ample
treatise on Tiigonometry, with tables of sines, tangents, and
secants, for every minute of the quadrant, to seven places of
decimals.

Ludolph van Ceulen, a Dutch mathematician, published,
about the year 1600, his well known treatise " De circulo etad-
scriptis," &,c. in which he treats of the properties of lines drawn
m and about a circle, and especially of chords, with the con-



TRIGONOMETRY. 48 1

struction of the canon of sines. Francis van Scliooten, in 1627
published at Amsterdam, a table of sines, tangents, and secants,
for every minute of the quadrant to seven places, which has
been esteemed a very accurate work.

These are the principal writers on Trigonometry, and the
tables of sines, tangents, and secants, before the change that
was made in the subject by die introduction of the logarithmic
calculus, which was first employed in this science about the
commencement of the seventeenth century, by the inventor,
the celebrated Baron Napier, of Merchiston, in Scotland, who,
in 1614, published his work entitled " Mirifici Logarithmorum
Canonis Descriptio," which contains the logarithms of numbers,
and the logarithmic sines, tangents, and secants, for every mi-
nute of the quadrant, together with the description and use of
the tables. But the person to whom we are chiefly indebted,
for the new and more advantageous form which the mode of
computation has since assumed, is Mr. Henry Briggs, at that
time professor of geometry in Gresham college, who has the
merit of having first proposed, both to the public in his lec-
tures, and to the illustrious inventor of the doctrine himself,
the improvement in the system of these numbers, which con-
sists in making the radix of the system 10, instead of 2.71828,
&c. as was done by Napier ; or which is the same thing, by
changing them from the hyperbolic to the present common,
or tabular logarithms. In 1624, Mr. Briggs published his
** Arithmetica Logarithmica," which contains the logarithms
of all the numbers from 1 to 20,000 and from 90,000 to
1 00,000 to fifteen places of figures. This table was completed
by Adrian Valcq, of Gouda, in Holland, who calculated the
seventy intermediate chiliads, and republished the " Arithme-
tica Logarithmetica," at that place, with these additional num-



Online LibraryWilliam ShepherdSystematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) → online text (page 39 of 44)