whole is an example of the value of those who have been born and educated in this country, as translators, interpreters, revenue and judicial officers, and public servants, wherever natives are concerned. Were such to perfect themselves in the native tongue and laws, and in the routine of British Administration, by taking advantage of the opportunities they enjoy above all others, whether native or English-born, they might distinguish themselves as a class far more than they now do. The preliminary Dissertation on Urdu, Hindi, Persian, and Arabic viewed philologically, and as "law languages," is good, and conveys not a little information that is new to the general student of languages.

An Elementary Treatise on Plane Geometry, according to the method of Rectilineal Co-ordinates. By the Rev. Thomas Smith, Missionary of the Free Church of Scotland, Calcutta. W. Whyte and Co.; Edinburgh. 1857.

This

Ir is not the practice of this Review to devote much attention to works, like that named above, of a purely scientific nature. has partially arisen from the small number, and the generally meagre quality, of such works that have been published in India. That under notice, independently of its scientific value, may be considered as having a claim on our attention, from the fact of its having been written by a gentleman who, till recently, conducted this Review, and contributed no small portion of its contents.

The Rev. Mr. Smith's work, though entitled "Plane Geometry," is not strictly limited to that department of mathematics which the name usually imports. The term "Plane Geometry" was given before the science had reached its full development, and when it was restricted to the properties of figures formed by the straight line and the circle; and in this confined sense it is most commonly employed. The title then fails adequately to convey a correct idea of the nature of this work, since it discusses not only some of the properties of rectilineal figures, and the circle, but also of the Conic Sections in general, and by means of the Cartesian system of coordinates. The author indeed seems to feel it necessary to defend the title, but the defence does not seem satisfactory, since the term employed is in common acceptation restricted to that branch of the science which forms but a portion of the substance of the work.

Of the intellectual exercise it has afforded, and the estimation in which it has been held by those who have cultivated and developed the science of pure geometry, some idea may be best formed from its history. Of its early dawn scarcely anything is known. Tradition points doubtfully to Egypt as the cradle of mathematical science, and as the source from which the first meagre knowledge of it was imported into Greece by Thales (639-548 B. C.); but from the admiration which, it is said, his easy applications of a few of its most elementary principles excited, it is evident that the

knowledge Thales did find among the Egyptians was of a very
limited character. Diogenes Laertius relates his measurment of the
height of a pyramid from its shadow, when that of a staff was equal
in length to itself, and that he discovered that the angle in a
semicircle is always a right angle. But for half a century there
were but few who devoted much attention to geometry or greatly
enriched it. Pythagoras (born. 580 B. C.) who also seems to have
visited Egypt, and was a man of decidedly mathematical cast of
mind, indefatigable in the pursuit of truth and knowledge, and
most grateful for it when found, devoted his attention to geometry,
and arrived at the well-known properties of triangles which form
the subjects of the 32nd and 47th propositions of Euclid's first book
of Elements. No results of comparatively equal value seem to have
been known at the time, as it is said his joy was very great on the
occasion.* That either of these philosophers imported any know-
ledge of geometry from Egypt has been doubted. Others, with
scarcely a shadow of reason, have supposed that the knowledge they
introduced was brought from India. Notwithstanding the boasted
antiquity of the science of this country, there seems to be very small
ground for supposing that what either of these men learnt on their
travels was so far fetched. It is only in later writers, when India
was to some extent known, that this origin is first hinted at; and
the hypothesis is attempted to be confirmed by the supposed resem-
blance of some of the tenets of Pythagoras to those of the
Gymnosophists of Hindustan. It is however very evident that the
whole Pythagorean system of philosophy bears the decided impress
of the Greek mind in its very genius. The little knowledge of the
fundamental ideas of mathematics which Thales and Pythagoras
taught, soon commended themselves to the Greek intellect, and if we
may trust to Suidas, in that early dawn Anaximander collected the
knowledge then existing into a systematic treatise. Correct principles.
being once attained, it needed little to entice the logical mind of the
Greek to deduce from them their most remote consequences, it was
indeed natural for it to do so. The problems which first enticed
the Greeks to the study seem to have been the quadrature of the
circle, the duplication of the cube and the trisection of an angle.
Whatever may have been the real origin of these difficult problems†,
they formed a lure, and the study proved so congenial to minds ever
striving after the most rigorous accuracy of thought and expression,
that it soon obtained the pre-eminence and the name of THE DISCI-
PLINE (μanos.) Till the time of Plato, the properties of figures
formed by the straight line and the circle were the subjects of geo-
metry, and direct deduction the only method of investigation.
new era in mathematics as well as in Philosophy was introduced by
the school of that eminent thinker. The disciples of the Platonic
school applied themselves in earnest to the problem of the duplication

* Ovid, Metam. xv.;
Jambl. de Vit. Pythag.
+ Pappus Col. Math.; Eratosthenes in Mesolabo.

Α

of the cube, or rather to that of finding two mean proportionals into which Hippocrates of Cos had resolved it, and this probably gave rise to the lines more particularly treated of in Mr. Smith's workthe Conic Sections, and of which we find the first notices in their applications by Menæchmus. It was in Plato's time, and probably in his school, that mathematical reasoning on account of its pre-eminent character received the title of the discipline, and by him a knowledge of the science was made an indispensible condition on the part of those admitted among the number of his pupils. He had felt their value, not in their applicability to the purposes of life, but to those of strengthening the power of vigorous deduction by frequent examples, and over which the conquest is sure if only the proper means be employed. Such was one principal means by which he developed the powers of such minds as Speusippus, Heracleides, Dinostratus, Menæchmus, and Aristotle. To Plato himself we are said to be indebted* for an invention which was to become a powerful instrument in the hands of his successors; this was geometrical analysis-a beautiful and ingenions "method of discovering truth by reasoning from things unknown, or propositions merely supposed, as if the one were given and the other were really true. A quantity that is unknown, is only to be found from the relations which it bears to quantities that are known. By reasoning on these relations, we come at last to some one so simple, that the thing sought is thereby determined. By this analytical process therefore, the thing required is discovered, and we are at the same time put in possession of an instrument by which new truths may be found out, and which, when skill in using it has been acquired by practice, may be applied to an unlimited extent. And a similar process enables us to discover the demonstrations of propositions supposed to be true, or if not true, to discover that they are false "t Geometrical analysis thus expresses a reversal of the order of the several steps in the demonstration of a theorem, or an examination of the conditions upon which a problem is to be constructed; and is thus distinguished by Leslie from Synthesis :-" Analysis presents the medium of invention; while Synthesis directs the course of instruction."

With the arms of Alexander, Geometry may be reckoned to have passed from Greece to take up its abode in the new capital he founded in Egypt, and to render its Museum famous in the history of Science. Ptolemy Lagus or Soter (323-284) founded the school of Alexandria, which flourished as an asylum of learned men, and the chief seat of science and literature for well nigh a thousand years. The first work that proceeded from this school, may, without challenge, take rank at the head of all human productions ;-a work which has been translated into the language of every nation that has made any considerable progress in science and civilization. In no science or department of human knowledge has a work appeared like the Elements of Euclid, which for two-thousand years has com

* Proclus in Euc.; Diog. Laert. in Vit. Plat.

† Playfair.

His

manded the admiration of the learned, and towards the improvement of which so little has been suggested. Of its author very little is known; the Orientals say he was a native of Tyre, but this is very doubtful, and the date of his birth can only be guessed at. Dedomena, or Book of Data, still exists, though considerably corrupted, and is the first in order of the books the ancients have left on Geometrical Analysis; it is a most valuable specimen of the rudiments of this branch. After Euclid arose the last two truly great mathematicians of antiquity, Apollonius Pergæus, and Archimedes. The former flourished at Alexandria about 210 B. C. and devoted his attention chiefly to the Conic Sections, and "the elegance and extent of his investigations of the most abstruse principles of this branch left but little to be added by more modern geometricians." In his own age he was known as the Great Geometer. Archimedes (287-212 B. C.) by means of the method of Exhaustions (employed by Euclid in his twelfth book of Elements) made great strides in geometry, discovered the quadrature of the parabola, and shewed that the circumference was more than 34, but less than 348 times its diameter. After these men the science they so succesfully cultivated for a long time made little advance. Diophantus in the second, Pappus in the fourth, and Proclus in the fifth century of our era, are almost the only mathematicians of note. With the decay of Greek power almost all scientific learning seems to have been neglected.

Among the Romans scarcely any attention was paid to geometry, and almost the only Latin names mentioned in connection with the subject are those of Cæsar, who is said to have intended promoting the cultivation of Geometry throughout the Roman dominions; and of Boëthius in the sixth century, who translated the first book of Euclid and wrote a work on Arithmetic,-these continued the standard text-books till Euclid was again introduced to the knowledge of Europe by the Arabs.

So far the ancients :-Among the Greeks, the science, so far as their methods could conduct them, was almost perfect, and it was only men like Gauss who could make any valuable addition to it. In Conics they had prepared for the use of Kepler and Newton in the investigation of the planetary orbits, an instrument as complete and powerful as was their language for the reception and preservation of Inspired Truth; and in preparing these, whilst seeking to cultivate their own reasoning faculties to the utmost, they fulfilled no mean mission. The Roman mind was too conservative to earn honors in such fields,-it was essentially practical. Its natural element was not science, it was public business, legal practice, government, legislation and war.

An age of thick darkness intervened, and when the modern era dawned in the thirteenth century, Algebra and Geometry were introduced together, ere long to combine and assist in solving the

* D'Herbelôt, v. 'Aklides' and 'Oclides'

+Young's Lectures on Natural Philosophy, &c. Lect. 20.

great problems of this physical universe. Among others who advanced these kindred sciences may be named Leonardo Bonacci, Tartaglia, Bombelli, Vieta, Roberval, Harriot, and Fermat. Bombelli had given examples of the applications of Algebra to Geometry; Vieta extended these, and Harriot (1560-1621) wrote the first treatise on Algebra in its modern form, a work of great genius and originality. To him we are indebted for the method of forming compound or adfected equations by the continued multiplication of so many simple ones. Among other things this seems early to have attracted the attention of the celebrated Descartes (1596— 1667) in whose hands Algebraic Geometry grew into a science. His Geometrie was published in 1637, six years after the publication of Harriot's Algebra, from which it is pretty generally believed that he borrowed many of his improvements.* The second of the three books of which this work consists is a treatise on Analytical Geometry, the first of its kind. A curve is supposed to be traced out by a series of points determined from two lines or axes either perpendicular or oblique to one another, by means of an equation called, the equation of the curve. This equation contains two quantities expressing the distances of a point from the respective axes, and which are variable, each value of the one quantity giving a corresponding value of the other variable. Besides the two variables, called the ordinates, there may be several others which remain constant. If any series of values then be supposed for one of the variables uniformly and continuously changing its magnitude, it is not difficult to conceive how values may be always found for the other such that the conditions of the equation shall be always satisfied; and the points determined by setting off from the axes, distances always representing the contemporaneous values of the variables will indicate the curve. By this felicitous application of equations of two unknown quantities to the genesis of curves made by Descartes, the science of Geometry was completely revolutionised. From the law of the description of any curve, its equation is deducible, and from it all the properties of the curve flow, hence the equation may be looked upon as a formula embracing all the characteristics of the curve, and from which they may be discovered by general methods applicable to all curves. In this the main difference between the Cartesian and the ancient Geometry consists-that, whereas the latter proceeded on no general methods, and the discovery of every property required a separate effort of the intellect, the new science went on general and fixed rules for the determination of the properties of all curves. The latter as a method of discovery is indisputably the more powerful and productive of results, whilst it is not to be denied that the old Geometry was the best as a mental discipline. The method of Descartes soon drew the attention of mathematicians, and so much engrossed it as to cause them to neglect to a great extent the ancient methods for the new. On the continent this was strikingly the case; and from the time of Pascal to the present it has been chiefly cul

*Wallis's Algebra, p. 198.

DEC., 1858.

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