One of the alterations made with this view, respects the Doctrine of Proportion, the method of treating which, as it is laid down in the fifth of EUCLID, has great advantages accompanied with considerable defects; of which, however, it must be observed, that the advantages are essential, and the defects only accidental. To explain the nature of the former requires. a more minute examination than is suited to this place, and must therefore be reserved for the Notes; but, in the mean time, it may be remarked, that no definition, except that of EUCLID, has ever been given, from which the properties of proportionals can be deduced by reasonings, which, at the same time that they are perfectly rigorous, are also simple and direct. As to the defects, the prolixness and obscurity that have so often been complained of in the fifth Book, they seem to arise chiefly from the nature of the language employed, which being no other than that of ordinary discourse, cannot express, without much tediousness and circumlocution, the relations of mathematical quantities, when taken in their utmost generality, and when no assistance can be received from diagrams. As it is plain that the concise language of Algebra is directly calculated to remedy this inconvenience, I have endeavoured to introduce it here, in a very simple form however, and without changing the nature of the reasoning, or departing in any thing from the rigour of geometrical demonstration. By this means, the steps of the reasoning which were before far separated, are brought near to one another, and the force of the whole is so clearly and directly perceived, that I am persuaded no more difficulty will be found in understanding the propositions of the fifth Book than those of any other of the Elements.

In the second Book, also, some algebraic signs have been introduced, for the sake of representing more readily the addition and subtraction of the rectangles on which the demonstrations depend. The use of such symbolical writing, in translating from an original, where no symbols are used, cannot, I think, be regarded as an unwarrantable liberty: for, if by that means the translation is not made into English, it is made into that universal language so much sought after in all the sciences, but destined, it would seem, to be enjoyed only by the mathematical.

The alterations above mentioned are the most material that have been attempted on the books of EUCLID. There are, however, a few others, which, though less considerable, it is hoped may in some degree facilitate the study of the Elements. Such are those made on the definitions in the first Book, and particularly on that of a straight line. A new axiom is also introduced in the room of the 12th, for the purpose of demonstrating more easily some of the properties of parallel lines. In the third Book, the remarks concerning the angles made by a straight line, and the circumference of a circle, are left out, as tending to perplex one who has advanced no farther than the elements of the science. Some propositions also have been added; but for a fuller detail concerning these changes, I must refer to the Notes, in which several of the more difficult, or more interesting subJects of Elementary Geometry are treated at considerable length.

COLLEGE OF EDINBURGH;

Dec. 1, 1813

1. Geometry is a science which has for its object the measurement of magnitudes.

Magnitudes may be considered under three dimensions,-length, breadth, height or thickness.

2. In Geometry there are several general terms or principles; such as, Definitions, Propositions, Axioms, Theorems, Problems, Lemmas, Scho liums, Corollaries, &c.

3. A Definition is the explication of any term or word in a science, showing the sense and meaning in which the term is employed.

Every definition ought to be clear, and expressed in words that are common and perfectly well understood.

4. An Axiom, or Maxim, is a self-evident proposition, requiring no formal demonstration to prove the truth of it; but is received and assented to as soon as mentioned.

Such as, the whole of any thing is greater than a part of it; or, the whole is equal to all its parts taken together; or, two quantities that are each of them equal to a third quantity, are equal to each other. 5. A Theorem is a demonstrative proposition; in which some property is asserted, and the truth of it required to be proved.

Thus, when it is said that the sum of the three angles of any plane triangle is equal to two right angles, this is called a Theorem; and the method of collecting the several arguments and proofs, and laying them together in proper order, by means of which the truth of the proposition becomes evident, is called a Demonstration.

6 A Direct Demonstration is that which concludes with the direct and ce tain proof of the proposition in hand.

It is also called Positive or Affirmative, and sometimes an Ostensive De monstration, because it is most satisfactory to the mind