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In working an exercise, the student must be careful to supply all the constituent parts of a Proposition. These are stated by Proclus to be as follows :
1. The Proposition or general enunciation, which states in general terms the conditions of the problem or theorem.
2. The exposition or particular enunciation, which exhibits the subject of the proposition in particular terms as a fact, and refers it to some diagram described.
3. The determination contains the predicate in particular terms, as it is pointed out in the diagram, and directs attention to the demonstration, by pronouncing the thing sought.
4. The construction applies the postulates to prepare the diagram for the demonstration.
5. The demonstration is the connection of syllogisms, which prove the truth or falsehood of the theorem, the possibility or impossibility of the problem in that particular case exhibited in the diagram.
6. The conclusion is merely the repetition of the general enunciation, wherein the predicate is asserted as a demonstrated truth.
Each deduction must be considered by itself, for no general rules can be given that will be found applicable to the solution of all kinds of exercises. The individual conditions of each exercise will supply the student with the suitable constructions, and the solution deduced therefrom will be adjudged possible or impossible, according to known principles consistent with the data of a problem or the hypothesis of a theorem.
In conclusion, the student must again be reminded that an intelligent and precise knowledge of the fundamental principles of Geometry must be gained, before the student can apply them intelligently and successfully. Unless these be mastered thoroughly before attempting the solution of deductions,-error, confusion, and failure will necessarily result. He must therefore strive to obtain an accurate and comprehensive knowledge of the Propositions of Euclid, their connection and mutual dependence one upon the other, before proceeding to work exercises bearing upon them.
The following remarks given by Ritchie will be found useful :
1. A point is said to be given, when its position is either given, or may be determined.
2. A line is given in position when its direction is given; in magnitude, when its length is given.
3. A line is given in position and magnitude, when both its direction and magnitude are given.
4. The position of a point can be found only-first, by a straight line cutting another straight line ; second, by a straight line cutting the circumference of a circle; or, third, by the intersection of the arcs of two circles.
5. The position of a line is found, when any two points in it are found; and its magnitude, when the extreme points are found.
IN study and pursuance of a notion, first work it out by thyself as far as thou canst, and make it lie as clear and distinct in thy head as possible ; and then (but not before) consult books and discourse with thy associates. For remember, thou art not always to live on reliance, and go in leading-strings.
DR. T. FULLER.