76G

THEORY AND PRACTICE OF ENGINEERING.

Book II.

Fig. 805.

Fig. 806.

to each other; so that the first shall bear the same proportion to the second that it does tothe third. A third proportional is requiredto the lines A B and C D, the first of whichis double the length of the latter ; draw anyangle less than a right angle, as E, F, G.

Carry the line A B from F to H on F G ;carry C D on F E from F to I, and draw theline 1 H ; then carry C D from H to K onthe line towards G; make L K parallel toI H, then IL is the third proportional re-quired.

When a fourth proportional is required ,an angle, as S F V, is set out at pleasure;from the point F, the line M N must be setout towards the point V, which will termi-nate at X. Carry the line O P on F S,v/here it will terminate at Y, draw the lineY X ; then carry Q R towards V, and it willend at Z. Draw IZ parallel to YX, andthe length YI will be the fourth proportional,as shown.

A mean proportional , found between thelines AB and KG , is found to be CD:for example, draw the line E P at pleasure,and set upon it the line A B and K G ,which will be found to terminate at H ;then divide EII into two equal parts bythe point I; from this point as a centre,describe a semicircle. From the point G,which answers to the length of the line A B,raise a perpendicular, which will cut thesemicircle at K; the line G K is the meanproportional to the line AB, CD: or theline A B is to K G , as K G is to C D. Shouldit be required to find a mean proportionalto the lines LM and OQ, draw the lineE P at pleasure, and carry on it the twolines L M, O Q. Describe a semicircle, andraise a perpendicular as before: then fromE as a centre, as with the radius E K, de-scribe an arc K R, which cuts the line E P inS ; then the line E S is a mean proportionalbetween the two given lines L M and O Q ;that is to say, L M is to E S, as E S is to O Q.

Arithmetical Proportion is when four mag-nitudes are proportionals ; A, B, C, D may

A C

represent them numerically : then = =r‘

B D

Three straight lines are in harmonicalprogression, when the first is to the third,as the difference of the first and second tothe difference of the second and third.

Pythagoras is said first to have noticed inchords of the same thickness and tension thesounds of the fifth and its octave. Theselengths are as 1, §, and the first of whichis to the third, as the difference of the firstand second is to the difference of the secondand third.

If a musical string is called C O, and itsparts D O, E O, F O, GO, A O, B O, CO,be in proportion to one another as the num-bers 1, §, $, 3, t ?, A, h their vibrations will

/

/

\

\

/

;s

S

E I C

1

R

H

I‘ ig. 807.

Fig. 808.

exhibit the system of eight sounds, which are expressed by the notes C, D, E, F, G, A, B, CHarmonical Proportion , as it relates to architecture, is that where three numbers are insuch relation, that the first is to the third, as the difference of the first and second, isto the difference of the second and third; thus 2, 3, and 6, are such numbers, because2:6;: 1:3; and four numbers are said to be in harmonical proportion when the first is