and are most frequently made use of by the ancient gometers Book VI. in the solution of other problems; and therefore are very ignorantly left out by Tacquet and Dechales in their editions of the Elements, who pretend that they are scarce of any use. The cases of these problems, wherein it is required to apply a rectangle which shall be equal to a given square, to a given straight line, either deficient or exceeding by a square ; as also to apply a rectangle which shall be equal to another given, to a given straight line, deficient or exceeding by a square, are very often made use of by geometers. And, on this account, it is thought proper, for the sake of beginners, to give their constructions as follows: 1. To apply a rectangle which shall be equal to a given square, to a given straight line, deficient by a square : but the given square must not be greater than that upon the half of the given line. Let AB be the given straight line, and let the square upon the given straight line C be that to which the rectangle to be applied must be equal, and this square by the determination, is not greater than that upon half of the straight line AB. Bisect AB in D, and if the square upon AD be equal to the square upon C, the thing required is done : but if it be not equal to it, AD must be greater than C, according L H K to the determination: draw DE at right angles to AB and make it equal to C: A B D G produce ED to F, so that EF be equal to AD or DB and from the centre E, at the distance EF, describe a C circle meeting AB in G, E and upon GB describe the square GBKH, and complete the rectangle AGHL; also join EG. And because AB is bisected in D, the rectangle AG GB together with the square of DG is equal a to (the square of DB, that is, of EF or EG, that is, a 5.2. to) the squares of ED, DG: take away the square of DG from each of these equals; therefore the remaining rectangle AG, GB, is equal to the square of ED, that is, of C: but the rectangle AG, GB is the rectangle AH, because GH is equal to GB; therefore the rectangle AH is equal to the given square upon the straight line C. Wherefore the rectangle AH, equal to the given square upon C, has been applied to the given Which was to Book VI. straight line AB, deficient by the square GK. be done. 2. To apply a rectangle which shall be equal to a given square, to a given straight line, exceeding by a square. E Let AB be the given straight line, and let the square upon the given straight line C be that to which the rectangle to be applied must be equal. Bisect AB in D, and draw BE at right angles to it, so that BE be equal to C; and having joined DE, from the centre D at the distance DE describe a circle meeting AB produced in G; upon BG describe the square BGHK, and complete the rectangle AGHL. And because AB is bisected in D, and produced L K to G, the rectangle AG, GB H together with the square of DB a 6.2. is equal a to (the square of DG or DE, that is, to) the squares F A D B G C 3. To apply a rectangle to a given straight line which shall be equal to a given rectangle, and be deficient by a square. But the given rectangle must not be greater than the square upon the half of the given straight line. Let AB be the given straight line, and let the given rectangle be that which is contained by the straight lines C, D which is not greater than the square upon the half of AB; it is required to apply to AB a rectangle equal to the rectangle C, D, deficient by a square. Draw AE, BF at right angles to AB, upon the same side of it, and make AE equal to C, and BF to D : join EF, and bisect it in G; and from the centre G, at the distance GE, describe a circle meeting AE again in H; join HF, and draw GK parallel to it, and GL parallel to AE, meeting AB in L. Because the angle EHF in a semicircle is equal to the right Book VI. angle EAB, AB and HF are parallels, and AH and BF are parallels ; wherefore AH is equal to BF, and the rectangle EA, AH equal to the rectangle EA, BF, that is, to the rectangle C, D: and because EG, GF are equal to one another, and AE, LG, BF parallels : therefore AL and LB are equal ; also EK is equal to KH a, and the rectangle C, D from the a 3. 3. determination, is not greater than the square of AL the half of AB; wherefore the rectangle EA, AH is not greater than the square of AL, that is of KG: add to each the square of KE; therefore the square b of AK is not greater than the b 6.2. squares of EK, KG, that is, than the squre of EG; and C consequently the straight line AK or GL is not greater E han GE. Now, if GE be equal to GL, the circle EHF K touches AB in L, and there. fore the square of AL is c c 36. 3. equal to the rectangle EA, H AH, that is, to the given rect M L N angle C, D; and that which A B was required is done : but if EG, GL be unequal, EG Q must be the greater : and therefore the circle EHF cuts the straight line AB ; let it cut it in the points M, N, and upon NB describe the square NBOP, and complete the rectangle ANPQ: because ML is equal to dd 3. 3. LN, and it has been proved that AL is equal to LB; therefore AM is equal to NB, and the rectangle AN, NB equal to the rectangle NA, AM, that is, to the rectangle e EA, AH, or the e Cor. 36. rectangle C, D: but the rectangle AN, NB is the rectangle 3. AP, because PN is equal to NB: therefore the rectangle AP is equal to the rectangle C, D; and the rectangle AP equal to the given rectangle C, D has been applied to the given straight line AB, deficient by the square BP. Which was to be done. 4. To apply a rectangle to a given straight line that shall be equal to a given rectangle, exceeding by a square Let AB be the given straight line, and the rectangle C, D the given rectangle, it is required to apply a rectangle to AB equal to CD, exceeding by a square. Draw AE, BF at right angles to AB, on the contrary sides of it, and make AE equal to C, and BF equal to D: join EF, and bisect it in G ; and from the centre G, at the distance P O Book VI. GE, describe a circle meeting AE again in H; join HF, and draw GL parallel to AE; E C. D G Q angle EAB, AB and HF 1 B H F to the rectangle EA,BF,that is, to the rectangle C,D: and because ML is equal to LN, and AL to LB, therefore MA is equal to BN, and the rectangle AN, NB to MA, AN, that is, a 35. 3. a to the rectangle EA, AH, or the rectangle C, D: therefore the rectangle AN, NB, that is, AP, is equal to the rectangle C, D, and to the given straight line AB the rectangle AP has been applied equal to the given rectangle C, D, exceeding by the square BP. Which was to be done. Willebrordus Snellius was the first, as far as I know, who gave these constructions of the 3d and 4th problems in his Appollonius Batavus: and afterwards the learned Dr. Halley gave them in the Scholium of the 18th prop. of the 8th book of Appolonius's conics restored by him. The id problem is otherwise enunciated thus: To cut a given straight line AB in the point N, so as to make the rectangle AN, NB equal to a given space: or, which is the same thing, having given AB the sum of the sides of a rectangle, and the magnitude of it being likewise given, to find its sides. And the fourth problem is the same with this, To find a point N in the given straight line AB produced, so as to make the rectangle AN, NB equal to a given space : or, which is the same thing, having given AB the difference of the sides of a rectangle, and the magnitude of it, to find the sides. Book VI. PROP. XXXI. B. VI. In the demonstration of this, the inversion of proportionals is twice neglected, and is now added, that the conclusion may be legitimately made by help of the 24th prop. of B. 5. as Clavius had done. PROP. XXXII. B. VI. The enunciation of the preceding 26th prop. is not general enough; because not only two similar parallelograms that have an angle common to both, are about the same diameter; but likewise two similar parallelograms that have vertically opposite angles, have their diameters in the same straight line: but there seems to have been another, and that a direct demonstration of these cases, to which this 32d proposition was needful : and the 32d may be otherwise and something more briefly demonstrated as follows: PROP. XXXII. B. VI. If two triangles which have two sides of the one, &c. Let GAF, HFC be two triangles which have two sides AG, A D H Н a 31.1. let it meet GF produced in K; because AG, KC are each of them parallel to FH, they are parallel b b 30.1. to one another, and therefore the B K C alternate angles AGF, FKC are cqual : and AG is to GF, as (FH to HC, that is c) CK to KF;c 34, 1, wherefore the triangles AGF, CKF are equiangular d, and the d 6.6. angle AFG equal to the angle CFK: but GFK is a straight line, therefore AF and FC are in a straight linee. e 14. 1. The 26th prop. is demonstrated from the 32d, as follows: If two similar and similarly placed parallelograms have an angle common to both, or vertically opposite angles; their diameters are in the same straight line. First, Let the parallelograms ABCD, AEFG have the angle BAD common to both, and be similar, and similarly placed ; ABCD, AEFG are about the same diameter. |