MECHANICS. 



expressed l-y p>int, or by inverting the fraotirm which exprewie* 



A ratio in said to be compounded of two otbr ration when it 

 to the product of th<> two ratio*. Thou, JJ is a ratio 

 .o ratios | nod {. 



i .airs of number* may hare tho same ratio. Thnn, 

 the ratios f , jj, $g, are all oquul. 



, of nnmbera hare the same ratio, the four 



:-( involved are noid to form a proportion; an 

 solves, in reference to tl. a subsisting among them, are 

 called jintiMirtioii.' 16, are proportional*, 



because tho ratio j, or 8 : 4 = the ratio }J, < 



A |.!-..|... :::: i fxpresBod i i ling tho sign of equality 



: ro o<iu:il rutiii , in- l>y placing four dot; m tin; 

 fui-ni of :i Ki|iimv, thus : : between them. 



Tims, tin- proportionality of U, 4, 12, 1G, might be expressed 

 in uiiy one of tin* throe following ways: 



{-*; 3:4-12:10; 3:4 "12:16. 



Th.> last expression would be read, 3 is to 4 aa 12 ia to 1(J. 



Tin- i ' and " terms of a proportion are called tho 



extremes ; thminiddlo two, the means. 



I, tlio product of the extremes 



! t'J til'' ir ' 



any proportion, 3 : 4 : ; 9 : 12, for instance. Expressing 

 ual form, we have J = $, and reducing these 

 fruitions to a common denominator 12 x 4, we get 



l *JLl = *JL, or 12 X 3 = 4 X 0. 



Now, 1 2 and 3 are the extremes, and 4 and 9 are the means, 

 of the given proportion. 



Conversely, if the product of two numbers is eqnal to the pro- 

 duct of any other two numbers, the four numbers will form a 

 proportion. Thus, since 



8X3-6X4 8, 4, 



or, 8:4 



Or we may write it thus, 8 : 6 



or, 3:6 



or, 4:8 



or, 4:3 



3 form a proportion ; 

 : 3 

 :3; 



9, 



e 



i 



4 



3:6; 

 :8:6. 



Thus we see that either product may be separated to form 

 tho extremes, and that, the order of either the means or the 

 extremes being interchanged, the numbers still form a pro- 

 portion. 



5. If three numbers bo given, a fourth can always be found 

 which will form a proportion with them. 



This is tho same thing as saying that if three terms of a pro- 

 portion be given, tho fourth can be found. 



Take any three numbers 3, 4, 5, for instance. Then we have 



3 : 4 : : 5 : fourth term. 



Therefore 



3 x fourth term = 5x4 (since tho products of the means and 

 extremes are equal). 



Therefore, dividing both of these equalities by 3 



5x4 

 Fourth term = 5 , the required number. 



a 



He-re we have found the fourth term, but we. could in tho 

 fiii ii ii' way find a number which would form a proportion with 

 the three given numbers when standing in any of tho terms. 

 For instance, for the second term wo should have 



and therefore 



3 : second term : : 4:5, 



4 x second term 5 x 3. 

 Hence, dividing both of these equalities by 4 



_ 6 x 3 



second term = 7 , 



and similarly for the other two term*. 



Tho most important application of proportion is the solution 

 of examples of this kind, whore three terms of a proportion ore 

 p-iven to find R fourth. This is what is usually called Rule of 

 which will be dealt with in a future lesson. 



i'.. It is evident that if the two terms of a ratio be multiplied 

 or divided by tho same quantity, the ratio is unaltered. 



Any set of numbers are said to be respectively proportional to 

 any other set containing tho same number when the one set can 

 be obtained from the other by multiplying or dividing all the 

 numbers of that set by the same number. Tims, 8, 4, 5 are 

 proportional respectively to 9, 12, 15, or to L }, f. 



7. To divide a givtn number into parti which tkatt be proper' 

 tional to any pfcwn number*. 



h given numbers together, and then, dividing the given 

 number into a number of parts equal to this sum, take as many 

 of these parts M are equal to the given numbers respectively . 

 EXAM rut. Divide 420 in proportion to the numbers 7, 5, 



:md :',. 



And therefore tho respectiYe parts are 



A x 400-1M. 

 A x 480 - 140. 

 A x 480- 84. 



These parts are evidently in the proportion of 7, 5, and 3, 

 and their sum, 1M + 140 -f 84 - 420. 



8. The same method will apply if the given number or 

 quantity is to bo divided proportionally to given fractions. 



EXAMPLE. Divide 266 into parts which shell be respeetfrerj 

 proportional to }, J, and f. 



Following exactly tbe same method M before, the 

 without reduction, would be 



,-T-f-, * ^nrn x ^-"rrfn * * 



Or we may proceed thus : 



Reducing tho fractions to their least cominc 

 which is 60, we get 



.',!, and }8. 



Now these fractions are proportional respectively to 40, 45, 48. 



Hence we have to divide 26G in the proportion of 40, 45, and 



48, to which tho required answer in, since 40 -f- 45 + 48 = 133, 



M x 866, to x 866, and A * , 



or 80, 90, and 90. 



EXERCISE 41. 



Find in their simplest form : 



1. The ratio of 14 to 7, 36 to 9, 8 to 32, 54 to . 



2. Tho ratio of 324 to 81, 802 to 99. 



3. The inverse ratio of 4 to 12, and of 49 to 6. 



4 Find the fourth term of the proportions, 3 : 5::6 '. ; 4 : 8'.:9 ' -j 



5. Insert the third term in the following proportions 3 : 5 :: - : 6; 

 4: 8::- :9; i : f :: - : ;. 



6. Insert the second term in tbe foDowing proportions 3 : : : 5 : 6 ; 

 4:-:: 8: 9; J : - .: \ : |. 



7. Insert the first term in the following proportions - : :: 5 : fl 



8. Find a fourth proportional to 2*13, '579, and 3'1-USO, 

 5 places of decimals. 



9. Divide 100 in the ratio of 3 to 7. 



10. Two numbers are in the ratio of 15 to 34, and the smaller is 75 ; 

 find the other. 



11. What two numbers are to each other ss 5 to 6, the greater of 

 them being 240 ? 



*#* As tests by which tho correctness of the processes of 

 addition, subtraction, multiplication, and division may be 

 ascertained, were given in Lessons in Arithmetic, H to V., it 

 has not been thought requisite to give answers to the Exercises 

 already given in attract Arithmetic. The answers will, how- 

 ever, be supplied to future examples in concrete Arithmetic. 



MECHANICS. IX. 



Tin STEELYARD. 



ANOTHER weighing instrument is the steelyard, which (Pig. 54) 

 is a lever of the first order, to tho short ann of which is attached 

 at 6 a hook from which the substance, w, to be weighed is 

 suspended, while on the long arm slides tho movable counter- 

 The object aimed at hi this instrument being that a 

 small weight, p, should balance a huge one. w, on the hook, it 

 ia clear that there must be a corresponding disproportion in the 

 arms the fulcrum, a, must be near one of the ends of the beam. 

 Further, since it is necessary that the steelyard should take an 

 horizontal position, both when loaded and unloaded at it* hook. 



