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William Shepherd.

Systematic 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

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of liberal and scientific education. Geometry literally sig-
nifies measuring the earth, or parts thereof; and it was pro-
bably first invented to enable people to ascertain their owii
property in land, since which it has been extended and ap-
plied to other things and for other purposes, insomuch that
geometry,, with arithmetic, is now regarded as the foundation
of all mathematics.

^' Geometry," says an excellent writer, " will enable a person
to think justly. Without it there is a certain method wanting
which is necessary to rectify our thoughts, to arrange our ideas,
and to determine our judgments aright. It is easy to perceive
in reading a book, even a moral one, whether the author be a
mathematician or not. I am seldom deceived in this obser-
yation. The famous French metaphysician would not have



458 MATHEMATICS.

composed the Inquiry after Truth * uor the famous Leibnitz
his Theodice, if they had not been mathematicians. We perceive
in their productions that geometrical order which brings their
reasonings into small compass, while it gives them energy and
method. Order is delightful ; there is nothing in nature but
what is stamped with it, and without it there could be no
harmony. We may likewise say that the mathematics are an
universal science, which connects all the rest, and displays
them in their happiest relations. Tlie mathematician, at the
^8t look, is sure to analyse and unravel a subject or propo-
sition with justness ; but a man who does not understand this
scieuce, sees only in a vague, and almost always in an im-
perfect manner. Apply yourself then to this great branch of
knowledge, so worthy of your curiosity, and so necessary to
the uses of life; but not in such a degree as to throw you
into absence ; endeavour to be always recollected, whatever
dire your studies. If I were young, and had leisure, I would
acquire a more extensive knowledge of geometry. I have
always cherished that scieuce M'ith a particular predilection.
My turn of mind made me seek with avidity every thing that
was methodical ; and I pay but little respect to those works
which are only the exercises of imagination. We have three
principal sciences, which I compare to the three essential
parts of the human composition : ^Theology, which, by its
spirituality, resembles our soul ; the mathematics, which, by
their combination and justness, express our reason ; and
natural philosophy, which, by its mechanical operations,
denotes our bodies ; aud these three sciences (which ought to
maintain a perfect harmony) while they keep within their
proper sphere, necessarily elevate us towards their Author, the
50urce and fulness of all light. Philosophy without geome-
tiy, is like medicine without chemistry. The greater number
of modern philosophers reason inconclusively, only because
they are unacquainted with geometry. They mistake sophisms

5Jakbrandu.



GEOMETRY. 459

for truths ; and if they lay down just principles, they deduce
false conclusions from them."

Herodotus, Diodorus, and Strabo, maintain that the Egyp-
tians were the first inventors of geometry ; and that the annual
inundations of the Nile were the occasions of ii; for that
river, bearing away all the bounds and landmarks of meA's
estates, the people were obliged to distinguish their lands by
.the consideration of their figure and quantity, and thus formed
for themselves a method or art which was the origin of geometry.
A farther contemplation of the figures of lands laid down for this
purpose, might probably lead them to the discovery of some of
the properties of those figures ; which speculation continually
improving," the art also improved till it laid claim to the rank,
of a science.

Geometry then, may be considered as the science of exten-
sion, or extended things, that is, of lines, superficies and
solids. Notwithstanding what has been said above of the
Egyptians being the inventors of Geometry, the fact has been
disputed, and the honour given, by very respectable authors,
to the Hebrews. There is, however, no doubt that the inhabit
tants of Egypt, in their ancient monarchical state, were ac-
quainted with the elements of geometry, though it does not
appear that they had gone deeply into it, since to Pythagoras,
who flourished about five hundred and tvyenty years before the
birth of Christ, and who had spent a considerable part of his
life in Egypt, was attributed the invention of certain proposi-
tions in Euclid, particularly the 47th of the first book, which is
called after his name, the " Pythagorean theorem," and for the
discovery of which he offered a hecatomb to the gods. Hence
it has been inferred, that the great learning of the Egyptians
was not geometrical.

From Egypt, geometry, probably in its infant state, passed
over into Greece; forThales, the Milesian, who flourished five
hundred and eighty-four years before Christ, was reported to be
the first of the Greeks, who, coming into Egypt, transferred
Geometry from thence into Greece. He is said to have disco-



460 MATHEMATICS.

vered several of tlie propositions of the first five books of the
Elements which go under the name of Euclid. After Thales,
came Pythagoras, already cited, who first of all abstracted geo-
metry from matter, and made many discoveries. Next flourished
Anaxagoras, Hippocrates, and many others, till we come to
Plato, than whom no one shed a greater lustre on the mathema-
tical sciences ; he made many considerable additions to geometry,
and upon the entrance to his academy, the inscription " Let no
one unacquainted with geometry enter," was written. The fifth
book of the Elenreuts is said to have been the production of
Eudoxus and to Aristeus, Isidore, and Hypsicles, we are in-
debted for the books of the solid geometry. After these Euclid
came, who collected the inventions and discoveries of others, dis-
posed them into order, in many respects improved tbem, and
left those Elements, by which, in some shape or other, the youth
of every succeeding generation^ from that time to this, have been
instructed in mathematics. Euclid died about two hundred
and eighty-four years before the birth of Christ.

The next to Euclid of the ancient writers, whose works are
extant, is Apollonius Pergeus, who flourished in the time of
Ptolemy Euergetes, about a century later than Euclid. The
third ancient Geometer, whose writings remain, is Archime-
des of Syracuse, who was celebrated at the same time with
Apollonius; to the works of this great man we shall have
occasion again to refer. We might mention many other names
of great celebrity among the Greeks, m hich have been immor-
talized by their skill in ancient Geometry. This people con-
tinued their attention to thS sciences, properly *so called, even
after they had been subdued by the Romans. Whereas
their conquerors were so little acquainted with this science,
even in the most flourishing time of their republic, that they
commonly gave the name of mathematicians to those who
pursued the chimeras of judicial astrology. Nor were they
more disposed to cultivate geometry, it will be readily ima-
gined, during the decline, and after the fall of the Roman
empire. The case was different with the Greeks, among



GEOMETRY. 46 1

V'hom Nve find many excellent geometers since tlie commence-
ment of the Christian era, and even after the translation of
the Roman empire. Ptolemy lived under Marcus Aurelius ;
and we are in possession of the works of Pappus of Alexan-
dria, who flourished in the time of Theodosius ; of the Com-
mentary of Eutocius, the Ascalonite, who lived in the middle
of the sixth century, on Archimedes' mensuration of a circle ;
and of ft Conmientary on Euclid by Proclus, who flourished
still later.

The inundation of ignorance and barbarism, to which We
have referred in a preceding chapter, was as unfavourable to
geometry as to the other sciences ; and the few, who even dared
to apply themselves to it, were calumniated as magicians. A
gleam of light, however, soon appeared, and in those times of
European darkness, the Arabians themselves, as we have seen,
became distinguished as the guardians and promoters of sci-
ence ; and from the ninth to the fourteenth century, they pro-
duced many astronomers, geometers, geographers, &c.; from
whom the mathematical sciences were again received into
Spain, Italy, and the other parts of Eur(>pe, at the close of the
fourteenth century. After this period, many editions of Euclid,
and many commentaries on his Elements were published.

At the revival of letters, there were few Europeans capable
of translating and commenting on the works of the ancient
Geometers, and the science of Geometry made little progress
till the time of Des Cartes-, who published his Geometrjfein
1637 ; but from tliat period to the present, it has abounded
with votaries in almost all civilized nations, but in none more
than in Great Britain.

The province of Geometry is almost infinite: few of our
ideas but may be represented to the imagination by lines, by
means of which they become of geometrical consideration : it
being geometry alone that makes comparisons, and finds the
relations of lines. Astronomy, music, mechanics, optics, and
in short all the sciences which consider things susceptible of
more or less, may be referred to geometry j for all speculative



462 MATHEMATICS.

truths, consisting only in the relations of things, may be referred
to lines. Consequences may be drawn from them, and these
consequences, again, being rendered sensible by lines, they be-
come permanent objects, which may be constantly exposed to
a rigorous attention and examination ; and thus we have op-
portunities both of inquiring into their certainty, and pursuing
them farther.

Before we proceed to speak of the elementary treatises on
this science, we may observe, that Geometry is founded on cer-
tain axioms, or self-evident truths, which none can deny, or
fail of understanding, and it is introduced by definitions of the
various objects which it contemplates, and the properties of
which it investigates and demonstrates ; such as points, lines,
angles, figures, surfaces and solids. Lines are considered as
straight or right, and curved ; and in their relation to one an-
other, either as inclined or parallel, or perpendicular to other
lines, or to certain given surfaces. Angles are considered either
as right, or acute, or obtuse ; or external, internal, vertical, &c.
Figures are investigated by. considering their boundaries, as
triangles, which in relation to their sides are equilateral, isos-
celes, and scalene ; and in reference to their angles, they are
either right-angled, obtuse-angled, and acute angled : as qua-
drilaterals, \\\\\c\\ comprehend the parallelogram, including
the rectangle and square, the rhombus and rhomboid, and the
trapesium ; as multilaterdl, including all other many-sided
figures ; as circles ; and as solids, including the prism, parallel-
epipedon, cube, pyramid, cylinder, cone, sphere, and the
frustra of the latter.

In enumerating some of the elementary treatises of Geome-
try, we shall observe, that to those persons who unite patient
attention with a true taste for the ancient method of demon-
stration, we may, notwithstanding what has been advanced bj
writers of considerable reputation, recommend as the most
useful and valuable work on this subject .

" The Elements of Euclid," viz. the first six books, together
with the eleventh and twelfth. The errors of Theon, and



GEOMETRY. 463

others, afe corrected, and some of Euclid's demonstrations are
restored. Also the book of Euclid's data, in like manner cor-
rected, by Robert Simson, M. D. Emerftus professor of ma-
thematics in the university of Glasgow,'' 8vo.

The Elements of Euclid, or, as they are frequently termed,
" The Elements," are divided into three parts, which respect
superficies, numbers, and solids. The first four books treat of
planes only ; the fifth, of the proportions of magnitudes in ge-
neral ; the sixth, of the proportion of plane figures ; the se-
venth, eighth, and ninth, give us the fundamental properties of
numbA's ; the tenth contains the theory of commensurable
and incommensurable lines and spaces ; the eleventh, and
the following four books, treat of the doctrine of solids. Of
these fifteen books, the first six, and the eleventh and twelfth,
are not only the most important, but those which are gene-
rally considered as the elementary parts of science.

Dr. Simson has added to the eight books of geometry,
Euclid's data, which is the first in order of the books that have
been written by the ancient geometricians, to facilitate and
promote the method of resolution and analysis. Some curious
and valuable geometrical notes are subjoined. In the third,
and, we believe, all the following editions, is a short but
masterly treatise oa Trigonometry, entitled " Elements of
plane and spherical Trigonometry." Tlie edition from which
we write, was printed at Edinburgh, and is uncommonly accu-
rate. There is, besides this, a beautiful edition in quarto, but
without the trigonometry, printed by Foulis.

Editions of the same work, by Cunn, Tacquet, Stone,
Whiston, and De Clfeles,- have considerable merit. The best
edition of De Cliales, is that published in 1748, by Mr.
Samuel Ashby.

To Cunn's edition is added, a treatise on the construction
of logarithms. Andrew Tacquet, a learned Jesuit, has' sub-
joined to his edition, select theorems from the works of the
great Archimedes. The editions by De Chales and Whiston



464 MATHEMATICS.

are chiefly valuable in having shewn the application of the Vft-
rious theorems in geometry to practical purposes.

EucUd's Elements', the whole fifteen books compendiously
demonstrated, witli Archimedes' tlieorems of the sphere and
cylinder, investigated by the method of indivisibles. By Isaac
Barrow, D. D. To which is annexed, Euclid's data, and a
brief treatise of the regular solids. By Thomas Haselden,
teacher of the mathematics, 8vo, 1 732.

Of this work the learned author shall speak for himself:
** My province," says Dr. Barrow, " was not that of writing
the Elements of Geometry as I pleased, but of demonstrating,
in as few words as possibly I could, the whole works of Eu-
clid. As to the seventh, eighth, ninth, and tenth books, although
they do not so nearly appertain to the elements of plain and
solid geometry, as the six preceding and two subsequent books,
yet none of the more skilful geometricians can be so ignorant
as not to know that they are very useful for geometrical mat-
ters, not only by reason of the affinity that there is between
arithmetic and geometry, but also for the knowledge of both
commensurable and incommensurable magnitudes, so exceed-
ingly necessary for the doctrine of plain and solid figifres.
Besides, I easily persuaded myself, tliat it would not be unac-
ceptable to any lover of these sciences, to have in his posses-
sion the whole Euclidean work."

This, and some of the editions before-mentioned, are only to
be purchased from second-hand catalogues ; but it frequently
happens, that a real treasure may be thus obtained at the ex-
pense of half-a-crown, or less.

We shall now direct the attention of 4he reader to some of
the abridgmeqts of " the Elements," of which, indeed, it
would be difficult to enumerate the half that have been pub-
lished from time to time. By many mathematicians, these
abridgments, usually known under the title of " Elements of
Geometry," are treated witli great contempt, as inducing un-
scientific notions in the learner's mind. By others, however,



GEOMETRY. 465

and those of no mean talents, it is thought high time to dis-
card Euclid's Elements, because they assert that science
cannot be exhibited in a more disgusting form than it is in
them. They farther contend, that " The Elements " are not,
by any means, necessary to lay a good foundation in mathema-
tics, and that few of the eminent mathematicians of Europe
have actually been initiated by the study of Euclid. " Non
nostrum inter hos tantas componere lites ;" but shall proceed
to mention some works of considerable and deserved repu-
tation :

" Elements of Geometry, with their application to the men-
suration of superficies and solids ; to the determination of the
maxima and minima of geometrical quantities ; and to the
construction of a great variety of geometrical problems." By
Thomas Simpson, F. R. S. 8vo.

The demonstrations made use of in this work are concise
and accurate. The treatise on the maxima and minima con-
sists of nineteen theorems, neatly demonstrated, and well
calculated to give the learner a proper taste for thisi part of
science. And the construction of the geometrical problems
is an useful application of the elementary books. The edition
before me, which I have read more than once, and which is
printed with tolerable accuracy, is the third, 1768 ; but there
have been several since that period.

" An Introduction to Geometry : containing the more useful
propositions of Euclid, and other authors. Demonstrated in
a clear and easy method, for the use^pf learners." By William
Payne.

Of this work, we can also speak with confidence, and will
venture to pronounce it a very plain and useful treatise : ex-
ceedingly well adapted to the capacity of learners, at a very
early age. With this introduction, we know, from repeated
experience, that children ten or twelve j'ears old, may, with-
out difficulty, become good geometricians. The first edition
was published in small quarto, 1767. The second is in
12mo, and was published in 1768. To this is added a neat

VOL. I. a H



466 MATHEMATICS.

introduction to mcDSuration. The quarto generally sells for
about 3s. 6d. and the other for 2s. 6d. but it is becoming
scarce, and is rising in value.

" Elements of Geometry ; containing the principal proposi-
tions in the first six, and the eleventh and twelfth books of
Euclid; with notes critical and explanatory." By John Bonny-
castle, of the Royal Military Academy, Woolwich, 8vo.

The number of propositions in this very excellent work, is
about one-fifth less than in Dr. Simson's Elements of Euclid.
Tlie enumerations and demonstrations are expressed in a more
concise and elegant manner, which, by persons not particularly
attached to the ancient mode of geometry, will be thought an
advantiige to learners five editions of it have been published.

" Geometry made Easy ; or a new and methodical expla-
nation of the Elements of Geometry. Containing a very easy
commentary on the first six, and the last five books of Euclid,
&c. &,c. &c. To which is added, an entire new, curious, and
exact method of exhibiting, in miniature, the various kinds
of solids, and their sections, by schemes cut out of paste-
boards," &c. By John Lodge Cowley. 8vo.

Mr.. Cowley makes use of the algebraic mode of reason-
ing : his propositions are neatly demonstrated, and the method
he has introduced of conveying an idea of the nature of the seve-
ral solids and sections of bodies, by pasteboard, folded up in the
forms of those solids, has, by persons of eminence, been thought
of considerable use in assisting the ideas of leaniers. Another
edition of this work, wi|b improvements, has been published
by Mr. Jones, Holborn.

An Appendix to the last mentioned work, 4to., besides tliat
of facilitating the study of Euclid's Elements, has this advan-
tage, that it is capable of shewing workmen how these
solids are to be made ; and in what manner they may be
divided, for the purpose of making models of them, by lines
drawn on paper.

" The Royal road to Geometry ; and familiar introduction
to the mathematics." By Thomas Malton. 8vo.



GEOMETRY*/' 467

This work comprehends every thing that is necessary for
the instruction of the learner in the theory and practice of
geometry. The first part contains the elements of geometry ;
the theory of mensuration ; and a demonstration of some
properties of ellipses. In the second, the author has inser-
ted the problems of Euclid's Elements, with others selec-
ted from various writers ; and an appendix on the construction
of ellipses ; proportional scales, and line of chords with prob-
lems illustrating their use.

** Elements of Geometry ; containing the first six books of
Euclid, with two books on the Geometry of Solids. To
which are added. Elements of plane and spherical Trigono-
metry." By John Play fair, F. R. S. Professor of Mathema-
matics in the university of Edinburgh. 8vo.

The four first books and the sixth are, with very few ex-
ceptions, the same as in the edition of Euclid, by Dr. Robert
Simson before-mentioned. In the fifth the author makes use of
an algebraic notation. The seventh book contains solid geome-
try, but not Euclid's. The rectification and quadrature of the
circle are discussed in the eighth book. Mr. Playfair has by
this work shewn himself an accurate writer and. an excellent
reasoner.

** Elements of Geometry, Geometrical Analysis, and Plane
Trigonometry," by John Leslie, F. R. S. E. This work, the
author informs us, is only part of a plan which he has in con-
templation, and which may be comprized in five volumes. Of
these, the second is intended to treat of the Geometry of
curve lines, the intersection of planes, and the properties of
solids, including the doctrine of the sphere, and the calcula-
tion of spherical triangles, with the elements of perspective
and projection. The third volume will be devoted to
Algebra, which is to be preceded by a short tract on the
principles of Arithmetic. The fourth and fifth volumes will
embrace the differential and integral calculus, with their prin-
cipal applications.

Mr. Leslie, in speaking of his reasons for introducing to
2 H 2



468 MArTlIEMATieS.

public notice, a new treatise of Geometry, says, " We shonlcf
form a wrong estimate, did we consider the Elements of
Euclid, with ail its merits, a finished production. That admi-
rable work was composed when geometry was making its
most rapid advances, and new prospects were opening on
every side. No wonder that its structure should appear loose
and defective. In adapting it to the actual state of the science,
I have, therefore, endeavoured carefully to retain the spirit of
the original, but have sought to enlarge the basis, and to dis-
pose the accumulated materials into a regular and compact
system. By simplifying the order of arrangement, I presume
that I have materially abridged the labour of the student. The
numerous additions which are incorporated in the text, so far
from retarding, will rather facilitate his progress, by render-
ing more continuous the chain of demonstration."

Mr. Leslie farther adds, that the view w hich he has given of
the nature of Proportion in the fifth book, w ill, he expects,
contribute to remove the chief difficulties attending that im-
portant subject. The sixth book, which exhibits the applica-
tion of the doctrine of ratios, contains a copious selection of
propositions, not only beautiful in them'selves, but which pave
the way to the higher branches of Geometry, or lead im-
mediately to valuable practical results.

In the part devoted to Geometrical Analysis, the first book
consists of a series of the choicest problems, rising above
each other in gradual succession. The second and third books
are almost wholly occupied with the researches of the ancient
Analysis. Of Mr. Leslie's Trigonometry, we shall speak
farther on. We have been thus particular in describing the
contents of Mr. Leslie's work, but after all, it does not
appear so well adapted to beginners as many of the others
which have been mentioned before ; and should, perhaps, be
considered rather as a second, than a first book for those who
are studying the science without the aid of an instructor.

One of the latest works of Geometry that have come with-
in our notice, is entitled " Geometria Legilinia, or an Elemen-



GEOMETRY. 466

tary system of Theoretical Geometry, adapted for the
general use of beginners in the mathematicqil Sciences," &c.
By Francis Reynard.

The author of this work assumes, that the Elements of
Euclid is a book not calculated for schools, and particulariy
unfit to be put into the hands of boys, who might well begin
a course of geometry, such as is here presented to them, at
the age of twelve. We so far agree with Mr. Reynard, that
we would not, in general, put Euclid into the hands of very
young persons, yet we must observe, tliat we have known it
taught with success in schools, and we have witnessed lads of
twelve or thirteen years of age, who have been instructed in



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 38 of 44)