A COURSE OF MATHEMATICS, &c. 1. QUAN GENERAL PRINCIPLES. UANTITY, or MAGNITUDE, is any thing that will admit of increase or decrease; or that is capable of any sort of calculation or mensuration : such as numbers, lines, space, time, motion, weight. 2. MATHEMATICS is the science which treats of all kinds of quantity whatever, that can be numbered or measured.That part which treats of numbering is called Arithmetic; and that which concerns measuring, or figured extension, is called Geometry.-These two, which are conversant about multitude and magnitude, being the foundation of all the other parts, are called Pure or Abstract Mathematics; because they investigate and demonstrate the properties of abstract numbers and magnitudes of all sorts. And when these two parts are applied to particular or practical subjects, they constitute the branches or parts called Mixed Mathematics.Mathematics is also distinguished into Speculative and Practical: viz. Speculative, when it is concerned in discovering properties and relations; and Practical, when applied to practice and real use concerning physical objects. Vol. I. 3. In 3. In Mathematics are several general terms of principles; such as Definitions, Axioms, Propositions, Theorems, Problems, Lemmas, Corollaries, Scholiums, &c. 4. 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. 5. A Proposition is something proposed to be proved, or something required to be done; and is accordingly either a Theorem or a Problem. 6. 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 triangle is equal to two right angles, this is a Theorem, the truth of which is demonstrated by Geometry. -A set or collection of such Theorems constitutes a Theory. 7. A Problem is a proposition or a question requiring something to be done; either to investigate some truth or property, or to perform some operation. As, to find out the quantity or sum of all the three angles of any triangle, or to draw one line perpendicular to another.- -A Limited Prob lem is that which has but one answer. An Unlimited Problem is that which has innumerable answers. And a Determinate Problem is that which has a certain number of an swers. 8. Solution of a Problem, is the method of finding the answer. A Numerical or Numeral Solution, is the answer given in numbers. A Geometrical Solution, is the answer given by the principles of Geometry. And a Mechanical Solution, is one which is gained by trials. 9. A Lemma is a preparatory proposition, laid down in order to shorten the demonstration of the main proposition which follows it. 10. A Corollary, or Consectary, is a consequence drawn immediately from some proposition or other premises. 11. A Scholium is a remark or observation made on some foregoing proposition or premises. Such 12. 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. 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 |