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flration of tliis theorem, in the Memoires de I'Acad. Roy. 

 an. 1693, p. 233, feq. Id. an. 1705, p. 397, feq. ; hut it 

 may he alfo demonftrnted pneumatically, thus : — Calling the 

 air remuining after the fail llroke, u\e firjl njsdttal ; that 

 after the fccond, the ftcoiul rijnluiil, &c. and remembering 

 that the air in the receiver is of the fame dcnfity as that in 

 the cylinder, when the pillon is raifcd ; it is evident, that 

 tlie quantity of air in the receiver, is to the quantity of air 

 in the cylinder, pipe, &c. as the capacity of the receiver to 

 tl\at of the cylinder, and confequcntly, the aggregate of 

 the air in the receiver and the cylinder. I.e. the whole primi- 

 tive air, is to the air in the veiTel alone, ;'. e. to the firil 

 refiduul air, as the aggregate of the capacity of the receiver 

 and tlie cylinJ^r, to the capacity of the receiver alone. 

 After the fame manner it may be proved, that the quantity 

 of the fidl relidual air, is to the fecond rcfidual, as the 

 aggregate of the capacity of the receiver and cylir.der to 

 the capacity of the veffel alone. And the fame proportion 

 does the fecond rcfidual bear to the third, and fo of the 

 reil. 



This may be ilhiflrated by an example. Siippofe the 

 capacity of the receiver to be twice as great as the capacity 

 of the cylinder or barrel, then will the capacity of tlrj 

 barrel he to that of the barrel and receiver together as one 

 to three ; and the quantity of air exhaufted at each turn of 

 the pump is to the quantity of air which was in the receiver 

 immediately before that turn, in the fame proportion. So that 

 by the tiril llroke of the pump, a third part of the air in the 

 receiver is taken away ; by the fecond llroke a tiiird part of 

 the remaining air is taken av.ay ; by the tliird ftmke a third 

 part of the next remainder is exhauiled ; and fu on con- 

 tinually ; the quantity of air evacuated at each ftroke de- 

 crcafing in the fame proportion with the quantity of air 

 remaining in the receiver immediately before that ilroke ; 

 for it is very evident that the third part, or any other de- 

 terminate part of any quantity muft be diminifhed in the 

 fame proportion with the whole quantity itfclf. And as the 

 quantity of air in the receiver is by each ftroke of the pump 

 diminiflied in the proportion of the capacity of the receiver 

 to the capacity of the barrel and receiver taken together ; 

 each remainder will therefore be always lefs than the pre- 

 ceding remainder in the fpme given ratio ; or, in other 

 words, thefe remainders will be in a geometrical progreflion 

 continually decreafing. To recur to the preceding example; 

 the quantity exhaufttd at the firft turn was a tlurd part of 

 the air in the receiver, and therefore the remaining air will 

 be tvvo-Uiirds of the fame ; and for the like reafon, the re- 

 mainder after the fecond turn will be two-thirds of the fore- 

 going remainder ; and fo on continually ; the deereafe being 

 always made, in the fam.e proportion of two to three ; con- 

 fequently the decreafing quantities themfelves are in a geo- 

 metrical progreflion. And as the quantities exhaufled at 

 every turn deereafe in the fame proportion with thefe re- 

 mainders ; therefore the quantities exiiaufled at every turn 

 are alfo in a geometrical progreflion. Thus it appears, that 

 the evacuations and the remainders do both deereaie in the 

 fame geometrical progreflion. If the remainders deereafe in 

 a geometrical progreflion, it is plain that, by continuing the 

 agitations of the pump, you may render them as fmall as 

 you pleafe ; that is, you may approach as near as you pleafe 

 to a perfeft vacuum ; but you can never entirely take away 

 the remainder. 



From the above reafoning it appears, that the produft 

 of the primitive air into the firft, fecond, third, fourth, &c. 

 refiduals, is to the product of the firft refidual into the 

 fecond, third, fourth, fifth, &c. as the product of the ca- 

 pacity of the receiver and cylinder together, multiplied as 



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often into itfelf as the number of ftrokcs of the pifton con- 

 tains units, is to the faftum arifing from the capacity of the 

 receiver alone, multiplied fo often by iti'elf ; that i";, as the 

 power of llie aggregate of the capacity of the receiver and 

 cyli^ider together, whofe exponent is the number of ftrokcs 

 of the pifton, to the capacity of the vefl'el alone, niifed fo 

 the lame power. ConfeqU'-ntly, the primitive air is to the 

 lall refidual, in the ratio of thofe jiowers. 



2. Tlie number of ftrokes of the pifton, together with 

 the capacity of the receiver and cylmder with the wire, &c. 

 being given ; to find the ratio of the primitive aifto the air 

 remaining. 



Subtract the logarithm of the capacity of the receiver, 

 from that of the finn of the cipaclly of the receiver and 

 the cylinder ; tiien, the remainder being multiplied by the 

 number of ftrokes of the pifton, the produc't will be a loga- 

 rithm, wluifc natural number (liews how often the primitive 

 air contains the remainder required. 



Thus, if tiie capacity of the receiver be 460, that of 

 the cyhnder 580, and the number of ftrokes of the pif- 

 ton 6 ; the primitive air will be found to the remaining air as 

 133,5 to I, or 1335 to 10. 



For, fuppofe llie capacity of the vefTel = -v, that of the 

 cylinder and veffel together := a, the number of ftrokes 

 of the pifton = «, and the remaining air = i. Since the 

 primitive is to the remaining air as a" to v'', the primitive 

 air will alfo be to the remaining air, as a" ■—■ V to I. Con- 

 fequcntly, if the remaining air be I, the logarithm of the 

 primitive air is log. a — log. -v x n. 



3. The capacity of the receiver and the barrel being 

 given ; to find the number of ftrokes of the pifton required 

 to rarefy the air to a given degree. 



Subtract the logarithm of the remaining air from the 

 logarithm of the primitive air; and the logarithm of the capa- 

 city of the receiver, from that of the aggregate of the ca- 

 pacity of the receiver and cylinder ; then, dividing the for- 

 mer difference by the latter, the quotient is the number of 

 ftrokes required. 



Let the primitive air be p, the remaining air r, and th.e 

 other quantities as before ; and we ftiail have ji : r : : a'' : ^'"; 

 and the log. p — log. r =z n X log. a — log. i\ ; and n = 

 log. p. — log. r -r- log. n — log. ^'. 



Thus, if the capacity of the cj-linder be fuppofed 5S0, 

 that of the receiver 460, and the primitive air to the re- 

 maining air, as 1335 to to: the .number of ftrokes required 

 will be found to be 6. 



4. The proportion of the primitive air to the remaining 

 air, together with the capacity of the receiver and the num- 

 ber of ftrokes of the pifton, being given ; to find the capa- 

 city of the barrel. 



Let the firft-mentioned proportion be that of ^ to r ; the 

 capacity of the receiver, %<, that of the barrel, .v, and the num- 

 ber of ftrokes of the pifton, n ; then p : r : : v-\-x\" : -0" ; 

 and log. p — log. r z= n x log. ij -j- .v — n X log. v : con- 



fequently, log. •y + - ■* ' -^ = log. t' -)- .v. Hence, 



n 



find the logarithm of the capacity of the receiver and bar- 

 rel, and from this the capacity itfelf, and fubtrafting 

 that of the receiver, the capacity of the barrel will be known. 

 For^ : /• : : 1335 : 10, t':=46o, andn=;6: confequcntly, 



/3. 1 25:6530 — i.cooocoo \ 



log. -v + x = 2.6627578 + (■' ^ "— ^ =) 



.3542755=3.0170333, the log. of 7040. Confequcnth-, 



.V = 1040— 460 = 5S0. See Wolf. Elem. Math. tom. ii. 



p. 2S9, Sec. Cotes's Ilyd. and Pneum. Leiflurcs, Iccl. T3. 



7 To 



