ABATER. 



Ufe of the Tab!:. — To render this theorem ufeful to thofe 

 who are not familiar witli the ufe of algebraic expreflions, 

 we Ihall give an example of the manner of calculating a 

 ftream of'water, all the operations being performed by com- 

 mon arithmetic, with the help of the preceding Table. 



1. If it is a ftream of water running in an uniform chan- 

 nel, take a fufficient number of dimenfions of the tranfverfe 

 feiftion of the channel, and by the rules of menfuration cal- 

 culate the area of its crofs feftion in fquare feet. Calculate 

 alfo, how much of the circumference of fuch crofs feftion is 

 touched by the water, not'mcluding its level top. 



Then divide the area in fquare feet by that portion of the 

 circumference in ,eet, in order to obtain the hydraulic mean 

 depth; this mull be multiplied by 12, to reduce it to inches. 

 Multiply the quotient by 4, and the refult is d, the number 

 which is to be fought in the firft column of the preceding 

 Table. . 



If it is a circular pipe of uniform bore, running full of 

 water, its internal diameter, taken in inches, is already 

 equal to four times the hydraulic mean depth, without any 

 computation ; and accordingly the diameter of the pipe in 

 inches is to be fought for in column i. 



2. By a fpirit-level or otherwife, afcertain the perpendi- 

 cular fall or difference of level, between any two diftant 

 points on the furface of the water, if it is an open ftream, 

 and find the diftance between thefe points of leveUing, by 

 meafuring upon a parallel to the furface of the ftream. 

 Thefe may be taken in any convenient meafures ; but the 

 fall and the diftance muft be reduced to the fame meafures : 

 then divide the fall by the diftance, and the quotient is /, or 

 a decimal number, which is ih^fme of the inclination of the 

 Itream. 



If it be a clofe pipe, the perpendicular fall muft be the dif- 

 ference of level between the furface of the refervoir and the 

 place of difcharge ; divide thia by the length of the pipe. 



3. Having found d, in column i of the Table, take out 

 the number oppofite to it in the fecond column, entitled 



— (that is, (/divided by a), and multiply this tabular num- 

 a 



ber by the decimal number s. 



Note It wiU fometimes happen that the exaft amount of 



d is not to be fouud in column I, but it will fall between two 

 of the numbers therein ; then take out the leaft of thofe num- 

 bers before d, and find how much is to be added thereto, by 

 the following rule : Take the difference of the two numbers 

 in col. 1. between which J falls ; alfo the difference of the 

 numbers oppofite to them in col. 2.; alfo take the difference 

 between the number d, and the leaft of the two numbers be- 

 tween which it falls. Now, by the Rule of Three, fay, as 

 the whole difference of the two numbers in col. i. is to 

 the fame in col. 2., fo is the difference between d and the 

 number above it in col. i. to a fourth number, which is 

 the proportional part to be added to the number of col. 2. 

 before d. 



4. Take out the tabular number from col. 3. which is 



c" 

 entitled — ; (that is, the fquare of c divided by the fquare 



of a). 



But here note, in cafe of calculating a proportional part, 

 (as direfted in the laft rule,) it is not always to be added (as 

 in col. 2.) ; but fometimes, on the contrary, it is to be fub- 

 trafted, accordingly as the numbers in that part of col. 3. 

 are increafing or decreafing ; and for greater eafe of difco- 

 vering this, a* is placed oppofite 14, and between 200 and 

 300 of col. I., to fhew the places where thefe changes take 

 place, from decreafe to increafe, and the contrary. 



10 



c. Multiply /, the refult of the ibcond operation, and — » j 



a j 

 the refult of the third operation, together, and to the pro- 



<:" 

 duft add —,, as found by the fourth operation : then extraft 

 a' 



the fquare root of this fum. 



6. Take out — from col. 4., and apply the proportional 

 a 



part as before, if neceffary ; deduft this number • — from 



et 



the fquare root laft found, and the remainder or refult is the 

 mean velocity of the ftream in inches per fecond, which was re- 

 quired. 



Should this refult be afterwards wanted in feet per minute, 

 the numbers lail obtained muft be multiplied by 60, and di- 

 vided by 12 ; or rather, multiplied at once by 5, which is 

 the fame thing. 



To obtain the quantity of water difcharged in a minute," 

 multiply the area of the feftion of the ftream by the velo- 

 city now found ; taking care, if the area is in fquare feet, 

 to exprefs the velocity of the water in feet ; or if the area is 

 in fquare inches, the velocity muft be expreffed in inches, 

 and the produft or refult will be in cubic feet or cubic 

 inches, accordingly. 



Example I . — The Academy of Sciences at Paris were oc- 

 cupied, during feveral months, with an examination of a 

 plan propofed by M. Parcieux, for bringing' the water of 

 Yevette into Paris ; and, after the moft mature confideration, 

 gave in a report of the quantity of water which M. De Par- 

 cieux's aqueduft would yield. Their report was afterwards 

 found erroneous in the proportion of at leaft 2 to 5 ; for 

 when the waters were brought in, they exceed the report in 

 this proportion. Indeed, long after the giving in the re- 

 port, M. Perronet, the moft celebrated engineer in France, 

 affirmed, that the dimenfions propofed were much greater 

 than were neceffary ; and faid that an aqueduft of ji feet 

 wide, and 3^ deep, with a flope of 15 inches in a thoufand 

 fathoms, would have a velocity of 12 or 13 inches per 

 fecond, and would bring all the water furniflied by the 

 propofed fources. The great diminution of expence occa- 

 fioned by the alteration, encouraged the community to under- 

 take the work. It was accordingly began, and partly exe- 

 cuted. The water was found to run with a veloeity of near i 

 19 inches, when it was 35 feet deep. 



M. Perronet founded his computation on his own expe- 

 rience alone, acknowledging that he had no theory to in- 

 ttruft him. 



Let us examine this cafe by our theorem. 



Firft, The area of the feftion is 3.5 feet deep x by 5.5 

 feet wide ;= 19.25 fquare feet. — The circumference which 

 the water touches, confiftsof the two fides of 3.5 feet each, 

 added to 5.5 feet, the bottom = 12.5 feet. The area 19.25 

 fquare feet divided by 12.5 feet gives 1.54 feet, for the hy- 

 draulic mean depth x 12 = 18.48 inches ; four times this 

 K d=. 73-92, which we are to feek in the firft column of ' 

 the table ; and may take 74. 



Secondly, To find s, take the fall 15 inches, or 1.33 feet, 

 and divide it by the diftance, 1000 fathoms, or 6000 feet ; 

 the refult is .00022, for s, or the fine of the incUnation. 



Take out from the table 

 the numbers correfpond- 

 ing to 74. 



We 



