opposite sides are respectively equal, does it follow that the figure must be a parallelogran ?

A. Yes. For if, in the last figure, you have the side CD equal to the side AB, and the side AC equal to the side BD; by drawing the diagonal BC, you have the triangle ABC with all three sides respectively equal to the three sides of the triangle CDB ; therefore, these two triangles must be equal; and the angle y, opposite to the side DB, must be equal to the angle w, opposite to the equal side AC; and the angle x opposite to the side AB, equal to the angle z, opposite to the equal side CD; that is, the alternate angles, y and w, x and z, are respectively equal : therefore, the side CD is parallel to the -side AB, and the side AC to the side BD (See page 31, 2dly); and the figure is a parallelogram.

Q. If in a quadrilateral there are but two sides equal and parallel, what will then be the name of the figure ?

A. It will still be a parallelogram. For if, in the last figure, the side CD is equal and parallel to AB, by drawing the diagonal CB, you have the two sides, CB and CD, in the triangle CDB, equal to the two sides, CB, AB, in the triangle ABC, each to each; and because the side CD is parallel to the side AB, the included angle y is equal to the included angle w; therefore, the two triangles must be equal (Query 1, Sect. II.), and the side AC is

also equal and parallel to the side DB, as before.


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If from one of the vertices of a rectilinear figure, diag

B onals are drawn to all the G other vertices, into how many triangles will this rectili- F near figure be divided ?

D A. Into as many, as the figure has sides less two. For it is evident, that if from the vertex A, for instance, you draw the diagonal AF, AE, AD, AC, to the vertices F, E, D, C, each of the two triangles AGF, ABC, will need for its formation two sides of the figure, and a diagonal; but then every remaining side of the figure will, together with two diagonals, form a triangle; therefore there will be as many triangles formed, as there are sides less the two, which are additionally employed in the formation of the two triangles AGF, ABC.

Q. And what is the sum of all the angles, BAG, AGF, GFE, FED, EDC, DCB, CBA, equal to?

A. To as many times two right angles as the figure ABCDEFG has sides less two. For as every rectilinear figure can be divided into as many triangles, as there are sides less two; and because the sum of the three angles in each tri

angle is equal to two right angles (Query 14, Sect. I.), there will be as many times two right angles in all the angles of your figure, as there are triangles; that is, twice as many right angles, as the figure has sides less two.





WHENEVER we compare two things with regard to their magnitude, and inquire how many times one is greater than the other, we determine the ratio, which these two things bear to each other. If in this way, we find out that the one is twice, three, four, &c., times greater than the other, we say that these things are in the ratio of one to two, to three, to four, &c. e. g. If you compare the fortunes of two persons, one

* It is the design of the author to give here à perfectly elementary theory of geometrical proportions, and to establish every principle geometrically, and by simple induction. Intending the above theory for those who have not yet acquired the least knowledge of Algebra, he is not allowed to identify the theory of proportions with that of algebraic equations (as it is done by some writers on Mathematicks), and then to find out the principles of the former by an analysis of the latter. There are several disadvantages inseparable from the algebraic method of considering a ratio as a fraction, besides the difficulty of making such a theory accessible to beginners. Neither ean an algebraic demonstration be made obvious to the eye like a geometrical one.

of whom is worth $10000, and the other $20000, you say, that their fortunes are in the ratio of one to two. Or if you compare two lines, one of which is two, and the other six feet long, you say of these lines, that they are in the ratio of one to three, because the second line is three times as long as the first.

It frequently occurs, that two things are to each other in the same ratio, in which two others are ; we then say, that these things are in proportion. This is frequently the case in the fine arts; but particularly in the science of Geometry, from which these proportions are called geometrical. To give an example : If you draw a house, you must draw it according to a certain scale ; that is, you will draw it one thousand, two thousand, three thousand, &c. times smaller than the building itself: but then you will be obliged to reduce every part of it in the same proportion. If, for instance, you draw the front of the house one thousand times smaller than the original, you must reduce the windows, doors, and every other part, in the same ratio. If, on the contrary, the windows were reduced two thousand times, whilst the doors and other parts were reduced only one thousand times, your picture would be out of proportion, because the different parts would be reduced by different ratios. In this case your

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