CHAMBERS'S INFORMATION FOR THE PEOPLE. 



or 



HK 



.'. HK is the fourth proportional required. 



PROP. XXXVIII. TV divide a given straight 

 line into any number of equal parts. 



Let AB be the given line which it is required 

 to divide into any num- 

 ber of equal parts. At 

 the point A draw a line 



AC, making any angle 

 CAB with the given 

 line. In this lay off the 

 equal portions AD, DE, 

 EF, &c. Join BF, and 



through E and D draw the lines EH and DG 

 parallel to BF. 



PROP. XXXIX. To describe a rectilineal figure 

 equal and similar to a given rectilineal figure. 



Let ABCDE be the given figure. Join AC, 



AD. Take the line FG equal to AB, and at the 

 points F and G in the line FG make the angles 

 GFH, FGH respectively equal to the angles BAG 

 and ABC. Again, at the points F and H, in the 



line FH, make the angles HFI, FHI equal re- 

 spectively to the angles CAD, DCA. At the 

 points F and I make the angles IFK, FIK equal 

 respectively to the angles DAE, ADE. Then the 

 figure FGH IK is evidently similar and equal to 

 the given figure ABCDE. 



PROP. XL. On a given straight line to describe 

 a figure similar to a given rectilineal figure. 



Let MN be the given straight line, upon which 

 it is required to describe a figure similar to the 

 given figure ABCDE. Join AC and AD. In 

 AB, or in AB produced, take Kb equal to MN, the 

 given line. Through b draw be parallel to BC, 



through c draw cd parallel to CD, and through d 

 draw de parallel to DE ; then it is evident that 

 the figure Kbcde is similar to the given figure 

 ABCDE. On MN, by the last proposition, de- 

 scribe the figure MNPQR equal and similar to 

 A.bcde. This is evidently the figure required. 



PROP. XLI. To find two lines having the same 

 ratio to each other as two given similar figures. 



Let ABC, DEF be the given similar figures ; it 

 is required to 'find two lines which shall have the 

 same ratio to each other as the figures have. To 



626 



BC, EF, corresponding sides in the two figures, 

 find the third proportional GH. Then the ratio 



of BC to GH is the same as the ratio of ABC 

 DEF. 



Since GH is a third proportional to BC and 

 EF, we have 



BC_ EF 

 EF GH 5 

 .'. BC GH - EF 2 . 

 B = BC 3 

 GH GH BC* 

 GH BC = EF 3 ; 



BC _ BC 3 

 '' GH ~ EF 2 ' 



But since the figures are similar, they are to 

 each other as 



BC 2 

 EF 25 



. BC _ ABC 

 " GH ~ DEF' 



Again, 

 But 



SOLID GEOMETRY. 



Definition I. A plane is a surface in which any 

 two points being taken, the straight line which 

 joins them lies wholly in that surface. 



Definition 2. A solid has length, breadth, and 

 thickness. 



THE DETERMINATION OF A PLANE. 



PROP. I. A plane is fixed or determined : 



(i.) If it pass through three points not in one 

 line. 



(2.) If it contain two lines which cut each other. 



(3.) If it contain two parallel lines. 



(i.) Let A, B, and C be the three points not in 

 one straight line. Let any plane whatever P pass 

 through A and B, then the line 

 AB lies wholly in that plane. 

 Let this plane revolve round 

 the line AB till it contains C, 

 then it is evident that in this 

 position the plane is fixed ; for, 

 if its revolution be continued, 

 it will no longer contain C. 

 Hence, of the infinite number of planes which al 

 pass through A and B, only one contains tl 

 point C. 



(2.) Let AB and AC be two lines cutting eac 

 other in A. Take any point B on the first line 

 and any point C on the 

 second, the three points B, 

 A, and C, which are not in 

 one line, determine a plane 

 P ; this plane contains the 

 given lines, and the plane 

 which contains the two lines 

 BA and CA contains the three points B, A, anc 

 C, and therefore coincides with the plane P. This 

 plane is therefore the plane of the two lines. 



