Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/293

Rh as it would lead us too far from Euclid's Elements of Geometry with which we are here occupied.

The first ten propositions in the second book of Euclid may be arranged and enunciated in various ways; we will briefly indicate this, but we do not consider it of any importance to distract the attention of a beginner with these diversities.

II. 1 and II. 3 are particular cases of II. i.

II. 4 is very important; the following particular case of it should be noticed; the square described on a straight line made up of two equal straight lines is equal to four times the square described on one of the two equal straight lines.

II. 5 and II. 6 may be included in one enunciation thus; the rectangle under the sum and difference of two straight lines is equal to the difference of the squares described on those straight lines; or thus, the rectangle contained by two straight lines together with the square described on half their difference, is equal to the square described on half their sum.

II. 7 may be enunciated thus; the square described on a straight line which is the difference of two other straight lines is less than the sum of the squares described on those straight lines by twice the rectangle contained by those straight lines. Then from this and II. 4, and the second Axiom, we infer that the square described on the sum of two straight Lines, and the square described on their difference, are together double of the sum of the squares described on the straight lines; and this enunciation includes both II. 9 and II.10, so that the demonstrations given of these propositions by Euclid might be superseded.

II. 8 coincides with the second form of enunciation which we liave given to II. 5 and II. 6, bearing in mind the particular case of II. 4 which we have noticed.

II. II. When the student is acquainted with the elements of Algebra he should notice that II. 11 gives a geometrical construction for the solution of a particular quadratic equation.

II. 12, II. 13. These are interesting in connexion with I. 47; and, as the student may see hereafter, they are of great importance in Trigonometry; they are however not required in any of the parts of Euclid's Elements which are usually read. The converse of I. 47 is proved in I. 48; and we can easily shew that converses of II. 12 and II. 13 are true.

Take the following, which is the converse of II. 12; if the square described on one side of a triangle be greater than the sum