4. All right-lined figures may be divided into triangles ;-then the Area of the figure the sum of the areas of the triangles: 41. I., Use 2. = 5. Lines from the centre of a regular polygon divide it into equal triangles; its Area therefore the Area of one triangle x the number of sides; the perpendicular x the perimeter: 41.I., Use 3. or, = 6. The Area of a trapezium is found, 40. I., Use 2, by taking half the sum of the parallel sides and the altitude, and multiplying the two quanti ties together. 7. In circles, measure the diameter D, and ascertain the circumference C; or measure the circumference, and ascertain the diameter; then, by 41. 3.1416 x D2 I., Use 4, the Area = 4 ; or = C2 4x3.1416 Dx C or= 4 ; or the By the use of other Geometrical Truths, the Student might have a much more extended view of the Principles which assist Mathematical Calculations;-but many of those truths lie out of the limits of an Elementary Work; and enough has been advanced to show the wide Application and Utility of the First and Second Books of Euclid's Plane Geometry. Indeed if any defence were required for confining the Examinations of Pupil Teachers, and of Scholars generally, in Elementary Schools, to the two books referred to, it is furnished by the very valuable Practical Results which have just been exhibited. Whoever has mastered and retains his familiarity with the Geometrical Principles now set before him, will possess sufficient knowledge of the subject for all the usual purposes of life,-and, what is more, will possess the means, whenever he chooses to employ them, of advancing with comparative facility to the higher and more abstruse mathematical learning. The right foundation has been laid; and the calls of professional duties and employments may be left to determine, whether the Student should remain satisfied with the mark attained, or go beyond it and labour in a wider field. If he is called, or prompted, to try the more difficult paths, he will never regret that his attention in youth was chiefly confined to the Introductory Books of Plane Geometry. It is the accuracy and the thoroughness of the early training,—and not the wide extent of the subjects, traversed indeed, but not known,-which increase the power of the mind; and the true aim of the Teacher is to strengthen power by a smaller quantity well done, than to waste it on a multitude of projects, to none of which it is able to do justice: The steam, that sounds a thousand jerking whistles, does not perform half so much useful work as that which keeps in steady motion a single loom. THE Principles of Plane Geometry, as taught in Euclid's Elements, are established, by proceeding in a regular series from Definitions, Postulates, and Axioms already known, to the consequences which flow from, and which are dependent upon, the Definitions, Postulates, and Axioms. This Method is entirely one of building up, or of putting together, and is therefore named Synthesis: "it commences with what is given, and ends with what is sought,"-the materials being furnished, out of them it fashions a garment; it takes elementary substances, and forms a compound. Analysis pursues an opposite course: it takes the compound, and resolves it into its constituent parts; the garment entire and completed is given for examination, and the aim is to discover of what it is made: Analysis begins with the thing sought, as a thing perfected and accomplished, and ends with whatever may have been supplied for the construction. No aids except those derived from Geometry were admitted by the Ancients in conducting an Analysis; and therefore the term Geometrical Analysis is employed. In analyzing a Problem, the solution is assumed to have been effected; and in analyzing a Theorem, the truth of the assertion contained in it is first of all admitted. When a Problem is analyzed, the object is-to discover something which, if done, would of necessity lead to the solution of the problem; and when a Theorem is analyzed, the object is-to determine whether the assertion is true or false. An example will more clearly show the difference between the analytical and the synthetical Methods: for this purpose we take the Problem In a given st. line, DE, to find a point, C, which shall be at the same distance from two other given points, A and B. By ANALYSIS.-We assume what was required, namely, that C, in DE, is the point equally distant from the given points A and B. = D F >B E The line CA = the line CB; and AB being joined, the figure ACB is an isosc. A. If now the line AB is bisected in F, we have, in the two triangles, CA CB, AF=BF, and FC common; .., by 8. I., ▲ AFC ▲ BFC, and AFC = 4BFC. But when two points, as A and B, are given, the line joining them AB, is given; and the line being given, its middle point F, is also given; at F, that middle point, a perpendicular, FC, is given; and consequently, if produced, its point of intersection, C, with the given line DE. But C was the required point, and the production of the perpendicular from F determines it. By SYNTHESIS.-Join the points A and B, and bisect AB in the point F; at F raise a perpendicular, and produce it to intersect DE in the point C ;that point C is equi-distant from A and B. = AFFB, FC common, and AFC BFC. ..the side CA= the side CB: which was required. In the Analysis, we have taken the Problem to pieces; in the Synthesis, we have put the parts together, and completed the purpose at which we aimed. It is by following similar Methods that other Geometrical Propositions may be analyzed and established. Analysis, however, is more suited for Problems ;Synthesis for Theorems. The Rules for conducting an Analysis are few;-for the mode of procedure depends in a great measure on the knowledge and skill of the Student; and the greater these are, the greater facility and clearness will he manifest in making an analysis. It is a process which calls forth all the resources of his mind,—and therefore a very improving exercise for the young Geometrician. The suggestions which are contained in RITCHIE's Geometry, p. 50, may be of service to the Learner, and therefore are recommended to his attention. "1st. As in every other study, endeavour to ascertain what it is you have to do; examine into the nature and meaning of the Proposition, and form a clear, well-defined idea of the quantities concerned in the investigation. 2nd. Construct every figure with exactness, that the eye may aid the judgment. 3rd. It will often be necessary to join certain points so as to form equal triangles, isosceles or equilateral triangles, and other right-lined figures; 4th. Often, also, to lengthen lines, to draw perpendiculars from certain points in a line, or to let fall perpendiculars from certain points on straight lines. 5th. Often, straight lines or angles must be bisected, one angle joined to another, so as to get the sum of two angles; or a line drawn within an angle, so as to get the difference of two angles. 6th. Also, it will often be necessary to draw lines parallel to certain lines through remarkable points, which may be either given or required. 7th. In short, the Learner must form such a combination of lines, angles, and circles, as will, in his judgment, lead to the discovery of the object required. If, after trial, he finds he cannot reach the required point, and take the citadel by the path he has sketched out, he must commence the attack anew by following a different road, and by adopting a different system of tactics." "REMARK.-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 st. line cutting another st. line; second, by a st. 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." A few EXAMPLES, selected from various sources, and restricted to the First and Second Books of the Elements, will show the Learner how an Analysis, or a Synthesis may be conducted. Ex. 1.—PROB.-Given an angle, A; a side, BC, opposite to it; and the sum, BD, of the other two sides of a triangle:-to construct the triangle. ANALYSIS.-The figure BAC is the required A, A the given angle, BC the side opposite to A, and BD the sum of the other two sides. Join DC: the sides of the triangle BA and A C the sum BD. Take away the common part BA, and the rem. B D SYNTHESIS.-At D, one extremity of BD, make an angle = ‹ BAC; and from B, the other extremity, draw BC, the given side, to meet DC in C; at C, make ACD = ADC, so that CA may meet BD in A: the triangle BAC will have its sides BA + AC BD the sum; its given 4, and its side = the given side. = = the |