ilANICS. 



Wl 



the denominators of whieh .. .power 



of li>. . ' 



\, jfo, ^{KJ, ^Jf, are Decimal Fract 



iro represented by a method of notation which 

 is an extension of that employed for whole numbers. 



In w!m!.- numbers tho figures increase in a tenfold ratio from 

 ; or, what is tho same thing, decrease in a tenfold 



right. If wo extend this method of r< ; 

 :o wanl-< tho right beyond tho units' place, any figure one 

 right of tho units' place will bo one-tenth of what 

 it" it were in tho units' place, and will thus really 

 :'>n ; any figure two places to tho right of 

 ice will bo one-hundredth of what its value would 

 in tho units' place; and so on for any number of 

 . -us. 



.vi! ehooso somo means of indicating tho point in 

 any row of figures at which tho units' place occurs, wo can 

 cimol fraction without tho trouble of express- 

 imal denominators. This is done by putting a dot, 

 1 1 orally called, between tho figure in 

 ':ico and the liirni-o in tho place to tho right of it, 

 which we may call tho tenths' place. Thus, 1'4 would mean 

 1 + * ; - 3 would mean ^; 3-14159 would mean 

 3 + A + T ;,, + 



2. Wo generally speak of any figure in a decimal as being in 



of decimals. Thus, in tho last example wo should 

 say that tho 5 is in tho fourth place of decimals, the 9 in tho 

 fifth place, and so on, reckoning from left to right. 



Observe that the denominator of tho fraction corresponding 

 to tho figure in any decimal place is unity followed by tho same 

 number of ciphers as tho decimal place'; or, what is tho some 

 thing, that the power of 10, which is tho denominator, is the 

 Bamo as the number of tho decimal place. 



3. The figures 1, 2, 3, 4, 5, 6, 7, 8, 9 in a decimal are some- 

 times called significant figures, or digits. Thus in such a decimal 

 as '0002356, we should say that 2 is tho first significant digit, 

 because it is the first figure which indicates a number, the 

 ciphers only serving to fix tho place in which the 2 occurs. 



4. To express a Decimal as a Vulgar Fraction. 



Qjt'7 ,.9 .1. < 4- _T 



**** ~ To Tuo ' To o ff 



Or (reducing the fractions to a common denominator, 1000) 

 300+40 + 7 



1000 1' 



0237 * f s + y-hs + fifoo + TooTTo-. 



Or (reducing the fractions to a common denominator, 10000) 

 _ + 200 + 30 + 7 _ 



10000 ~ iu"' 



Again 43-25037 = 43 + + & + r o + loSoo + ^ 

 Or (reducing tho fractions to a common denominator, 100000) 

 _ 4300000 + 20000 + 5000 + 0x 100 + 30 + 7 



100000 



Hence we see the truth of tho following 

 Rule for expressing a Decimal as a Vulgar Fraction. 

 Write down the figures which compose the decimal (both 

 integral and decimal part, if there is an integral part) for the 

 numerator, omitting the decimal point ; and for tho denominator 

 put 1, followed by as many ciphers as there are decimal places 

 in the given decimal. 



5. Conversely, if we have a fraction with any power of 10 for 

 its denominator, wo can express it as a decimal by placing a 

 decimal point before as many right-hand figures in tho nu- 

 merator as thcro are ciphers in tho denominator. Thus 



53459. 



If the figures in tho numerator be fewer than tho ciphers in the 

 denominator, we must place before the left-hand figure of tho 

 numerator ciphers equal in number to the excess of tho number 

 of ciphers in the denominator over tho number of figures in tho 

 numerator, and then prefix tho decimal point. For example 



rtWsv - '00235. 



06s. It will bo perceived from tho foregoing remarks that 

 placing ciphers on the right of a decimal does not niter its value, 

 for this does not alter the place of any of the figures. 



Thus, -23, -230, -2300 are all equal in value, for, expressed as 

 fractions, they are respectively &, *J, flgg. But prefixing 



cipher* between the decimal point and the first significant 

 figure doe* alter the value of the decimal, beeatue this alien 



th.i pi*.-,.* u f th, bi-i.ili.-ant .li-it*. Tin* -_':;, .::. -IM.J-; 

 all different values, being respectively equal to & jjfo, 



MECHANICS. V. 



PARALLEL FORCES. CENTRE OF GRAVITY. 



BEFORE proceeding to tho subject of the Centre of Gravity, I 

 muut direct your attention to two consequence* which flow 

 directly from the principle* establiahed in the but 1ta*on, and 

 furnish tho basis on which the properties of that centre rent 

 You have seen there that tho centre of a system of parallel 

 forces ia found by cutting in succession certain line* which join 

 certain points in certain definite proportions, namely, inversely 

 as tho forces acting at their extremities. Now, snob cutting can 

 give for each line, and therefore for all, as final result, only one 

 point. For example, tho centra of two parallel forces of six and 

 four pounds acting at two points, A B, of a body, a* in the last 

 lesson, is got by dividing A D into ten parts, and counting off 

 four parts next to A, or six to B, and the result evidently can be 

 only one point. If wo now suppose a third parallel force of 

 five pounds added, acting at some other point, c, of the body, 

 and join the point lost found with c, and divide the joining K*M 

 into fifteen parts, taking ten next to c, hen again only one 

 point is the result. And so on for any number of forces it can 

 be shown that there is but one centre. 



But, lest it should be thought possible that, on cutting these 

 lines in a different order of tho points, ABC, etc., a second 

 centre should turn up, let UH think that possible, and apply 

 forces at these points parallel to each other, but not parallel to 

 the lime joining these two centres. Their resultant then passes 

 through both of these points, and therefore must act in tho line 

 joining them, which is impossible ; since, as I have proved, it 

 must bo parallel to its components. 



Furthermore, you will observe that all these lines are cut 

 only in reference to the maynitudes of the forces ; no account is 

 taken of their direction. Whether they pull upwards or down- 

 wards, or obliquely to left or to right, so long as the magnitudes 

 remain the same, or even keep the same proportion say that of 

 six, four, and five the centre cannot change. Of course, tho 

 points are supposed not to change. Whatever be the number 

 of points and forces this is true ; as for three, so for any other 

 number. And mark, moreover, that it makes no difference how 

 this change of direction is produced, whether, leaving the body 

 in one fixed position, you make tho forces change directions as at 

 a and b (Fig. 17), or, preserving the direction, yon turn the body 

 round, as from a to c in the same Fig. In neither case does the 

 centre change. These results may be summed up in tho two 

 following propositions : 



1. A System of Parallel Forces acting at given points in a 

 body, has ONE Centre of Parallel Forces, and only one. 



2. Tho Centre of Parallel Forces does not change its position 

 when tho direction of tho forces is changed in reference to the 

 body. 



THE CENTRE OP GRAVITY. 



The centre of gravity is the particular case of the centre we 

 have been last considering, in which the forces are those by 

 which bodies on the earth's surface are drawn by attraction 

 towards its centre. The smallest body, particle, or atom, ia 

 drawn in proportion to its mass, equally with fhe largest ; and 

 it is to the tendency of these bodies so to move downwards in 

 obedience to this attraction, that we give the name of "weight." 

 The term "gravity," carries a similar meaning, being derived 

 from tho Latin gravis, heavy. 



Now, since every particle of matter is thus drawn to the 

 earth's centre, it is evident that the weight of all largo maimp ( 

 such as of a block of marble, beam of timber, or girder of iron, 

 is the joint effect, or the resultant, of the attractions of the sepa- 

 rate atoms. But these attractions are all so many parallel 

 forces ; for, pulling, as they do, towards the earth's centre, 

 which is nearly 4,000 miles away down in tho ground, they in- 

 cline, even in the largest objects, so little towards one another 

 that practically they may be considered not to meet, that is, to 

 bo parallel. Hence yon see that all the principles we h.ivo 

 proved about parallel forces hold good of the earth's attraction 

 of these atoms, and that wo may affirm that 



