Entered according to the Act of Congress, in the year One Thousand Eight Hundred and Forty-five, by W. E. DEAN, in the Clerk's Of fice of the Bouthern District of New-York.

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

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, show ing 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 forinal 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. E. 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