is equal to the squares of the two sides; .. the square described on the hypotenuse as a diameter, is equal to the squares described on the other two sides as diameters. A SUPPLEMENT TO THE ELEMENTS OF EUCLID. BOOK II. PROP. I. 1. THEOREM. If two given straight lines be divided, each into any number of parts, the rectangle contained by the two straight lines, is equal to the rectangles contained by the several parts of the one and the several parts of the other. any parts in the points E, F, and let the given straight line CD be divided first into two parts in the point G: The rectangle contained by AB and CD is equal to rectangles contained by AE and CG, by EF and CG, by FB and CG, by AE and GD, by EF and GD, and by FB and GD, taken together. From the point A draw (E. 11. 1.) AX 1 to AB; from AX cut off (E. 3. 1.) AI CG, and from IX cut off IH = GD, so that AH = CD; through I and H draw (E. 31. 1.) IN and HK parallel to AB, and through B, F, E, draw BK, FM, EL, parallel to AH: Then (E. 1.2.) the rectangle AN is equal to the rectangles contained by AE and CG, by EF and CG, and by FB and CG; also the rectangle IK is equal to the rectangles contained by HL and GD, by LM and GD, and by MK and GD; but (E. 34. 1.) HL= AE; LM = EF; and MK = FB; .. the rectangle IK is equal to the rectangles contained by AE and GD, by EF and GD, and by FB and GD; but the two rectangles AN and IK make up the rectangle AK, which is contained by AB and AH or CD; .. the rectangle contained by AB and CD is equal to the rectangles contained by AE and CG, by EF and CG, by FB and CG, by AE and GD, by EF and GD, and by FB and GD, taken together.. And, in the same manner, the proposition may be proved to be true, when the given straight line CD is divided into more than two parts. 2. COR. If the parts EF, FB, &c., into which AB is divided, and the parts CG, GD, &c., into which CD is divided, be each of them equal to AE, it is manifest that the rectangle contained by AB and CD is equal to the square of AE taken as often as is indicated by the product of the number of equal parts in AB, multiplied by the number of equal parts in CD. PROP. II. S. THEOREM. If a straight line be divided into two unequal parts, in two different points, the rectangle contained by the two parts, which are the greatest and the least, is less than the rectangle contained by the other two parts; the squares of the two former parts, together, are greater than the squares of the two latter, taken together; and the difference between the squares of the former and the squares of the latter, is the double of the difference between the two rectangles. A Let the given straight line AB be divided into K CD B two unequal parts, in the point C, and also in the point D: Then AD X DB < ACX CB; but AD2 + DB2 > AC+ BC; and the excess of AD2 + DB above AC+ CB' is the double of the excess of ACXCB above ADXDB. 2 For, bisect (E. 10. 1.) AB in K: Therefore, ACXCB+CK2 = AK2 and ADXDB+DK'= AK'S (E. 5. 2.) But CK2<DK'; ... AD×DB<ACXCB. Again, because AD2+DB2+2AD×DB=AB2 AC2+CB2+2ACX CB= AB2 (E. 4. 2.) and that, as hath been shewn ADXDB< ACXCB, ··. AD2+ DB2 > AC2+CB3. Lastly, since AD*+DB2+2ADXDB =AC2+CB2+2AC×CB, it is manifest, if from these equals there be taken AC2+CB2+>2ADXDB, that the excess of AD2+DB2 above AC+CB is the double of the excess of ACXCB above ADXDB. 4. THEOREM. PROP. III. In any isosceles triangle, if a straight line be drawn from the vertex to any point in the base, the square upon this line, together with the rectangle contained by the segments of the base, is equal to the square upon either of the equal sides. Let ABC be an isosceles A, and let AQ, be B 0 . D |