MENSURATION. ' ’ * 119 
The preceding Table exhibits, at one view, all the pro¬ 
portions of diameters, circumferences, areas of great cir¬ 
cles-, Superficial and folid contents, of thirty-nine leledt 
fpheres, and as many cubes of fimilar diameters, from the 
diameter i to 7964-, (and which number, if taken as in¬ 
dicative of miles, will be about the earth’s diameter;) 
ihowing wherein the peripheries, fuperficies, and folids, 
approximate to, agree with, or recede from, each other, 
and how much; aifo, how many times one is involved 
in the other, both under and over (the concurring num¬ 
ber) 6, with fimilar co-incidents, refpebting circumfe¬ 
rences and fuperficies ; and rules for discovering this 
eflential datum, d ■priori, never before exhibited. Alfo, 
the areas of their correspondent Squares, Superficies of 
Simultaneous hexahedra, and Solid contents of co-relative 
cubes, illuftrative of the harmony of numbers and the 
reciprocal proportions of all geometric bodies. 
As all circles are to each other as the Squares of their 
diameters, (i. e. the areas of circles,) and the Solidity of 
fpheres as the cubes of their diameters ; if thefe numbers 
of diameters in the table be confidered as indicative of 
inches, and we w'ifh to know the contents in feet, we 
mult divide the numbers of the areas and fuperficies by 
144, and the Solid contents by 1728. The quotient will 
be the anfwer, in Superficial or cubic feet. But, if we 
wifh to know their contents in lines, or Smaller parts, 
ive muff multiply the areas and fuperficies by 144, and 
the folid contents by 1728, to find their value; and So 
on for any other lower denomination, as thefe products 
will be the anfwer. But the fhorteft way, is to name 
them by the greateft denomination they will form inte¬ 
gers of 5 and proceed with the parts, if any remain, by a 
iecond multiplication of 144 and 1728, to find their ul¬ 
timate value. 
Again, as all diameters of cubes or hexahedra, as well 
as fpheres, under 6, produce a greater Superficies than 
Solid measurement; if we wifh to know how many times 
Such Solids are involved in their Superficial contents, it 
may be done as follows : Rule for all numbers not ex¬ 
ceeding fix ; Multiply the given diameter by fix, anrl 
divide by the Square of that diameter, thus; 
. r XS=6 -Hi ]i 2 =6 
r 25 X 6 = 7’5 
*■=[1.562 
5]i-25 2 =2 + -8 
i'S X6:2=9- 
-HA - 2 5 
] 1 ' 5 2 =+’ 
J75X6=io-5-HE°625]i-75—3-4285714 
2‘ X6=:i2- 
HU’ 
] 2 2 —3 
3 - X'6=i8- 
-h 9 - 
b 2 =*- 
4- X 62=24' 
-Hi6- 
]+ 2 =i ‘5 
5- X 62=30- 
-=[ 2 5 
b 2 =i’* 
6- X 62=36- 
-H 36 
]6 2 =21 
Rule II. If the number be more than fix, divide the 
given number or diameter by fix; the quotient will 
fhow the number of times the Superficies is contained in 
the folid; and, where there is an exaft proportion, the 
anfwer is given (without a remainder) either in whole 
numbers or conclufive decimals. The preceding exam¬ 
ples Show, that the very reverfe of this order takes place 
when the number is lefs than fix, and more fully So in 
that column of the table headed “ Excefs of Superficies 
to Solid,” which only obtains from one to fix, after which 
the Rule is inverted into “Excefs of Solid to Superficies,” 
while the number of involutions of circumferences in 
their fuperficies is always identical with the number 
t hofen for the diameter. 
It may be worthy of notice to remark, before we lole 
Sight of this interefting Subject, (of how many times the 
Solid is involved in the fuperficies, or, e emtrario, the 
fuperficies is contained in the Solid,) that the number of 
involutions, in both cafes, is in the inverfe ratio to the 
proximity to fix, in which number they both precisely 
agree ; becaufe, a hexahedron is fix times the area of one 
of its Squares; and a circle, whofe diameter is one, has 
■7854, or a fourth part of 3-1416, “the Superficies of its 
corresponding Sphere, for its area, and ’5236, or only a 
Sixth part of 3-1416, for its folid or cube measurement. 
Therefore, when the diameter comes to be fix times 
as great as at firft, by a Series of cubing, it recovers 
itfelffrom this Seeming folecifm, which only appears ano¬ 
malous by our method of notation, as would be eafily 
proved only by changing the denomination of our num¬ 
ber; for every Sphere bears the lame proportion to its 
parts and predicaments, whether great or Small. It mult 
be obvious to every one in the lealt acquainted with the 
menfuration of thefe bodies, that the advantage of this 
invention is immenle, in cafes where we only want to 
know what proportion the fuperficies bear to the Solid, 
or how many times one dimenfion involves the other or 
is involved in it. For inftance, let it be propofed how 
many times the fuperficies of Such a Sphere as our earth 
(admitting it to be 7964 miles diameter and a perfeft 
Sphere) is involved in its folid contents. By the time it 
has been Squared, cubed, multiplied by the cubing fa <51 or, 
and its fuperficies prepared by nearly a Similar operation, 
to form a proper divifor, and we have operated by this 
divilor, So as to obtain a true and fatisfaftory anfwer by 
the regular rules of arithmetic, we Should have calcu¬ 
lated four hundred figures, and the Sum not proved after 
all ; and we all know that the danger of error would be 
in direft proportion to the labour of the operation. Blit, 
by this invention, the fame refult is given by only 11 
figures, thus 6)7964(13271-. Befides, the table poiSefles 
the advantage of pointing out, in an infant, the collateral, 
as w-ell as direft, predicaments of all manner of fpheres 
and cubes. To the fra&ional diameters are added the 
duodecimal calculations as well, which, like the decimal’s, 
are extended to the Sixth or Seventh power, on purpofe to 
put the rule to the utmoft teft. In all the other cafes 
they are limited to four decimals only. But, whether we 
divide by the real products, or by their indices, the quo¬ 
tients are precisely the fame, if we-extend the numbers to 
infinity. 
To preferve uniformity in the table as much as pof- 
fible, the three -fractional diameters of ij, if, and if, Or 
1.25, 1.5, and 1.75, which are calculated duodecimally, 
are carried (for thole eftimates) to the bottom, and per- 
feCfly agree with the fame done decimally at top. And, 
as all decimals of feet and parts of feet, w-hen put into 
real practice, muft ultimately be reduced to duodecimals, 
the following rule cannot fail of being interefting to all 
pcrfons unacquainted with the method of doing it. The 
duodecimal worth of any other Sum in the table of cir¬ 
cles or Spheres may be readily found and exhibited, in 
the duodecimal manner, by perfons who are totally un¬ 
acquainted with the nature of decimal calculation, only 
by calling the whole numbers (i. e. the figures pricked off 
towards the left hand) integers of the dimenfion Sought, 
(fay feet,) and then Submitting the remaining fractions to 
a continual multiplication by twelve, observing at every 
operation to prick off as many decimal figures, on the 
right hand, as firft ufed, calling thofe on the left, parts 
or primes to the integer; then, again, multiplying the 
remainder, (omitting every time the Sum pricked off,) 
when .the rejebted figure or figures will be Seconds ; and 
lo on, until all the decimals are either evolved or lb 
weakened as to be beneath notice. Thus, for the area 
of 36 we find the tabular Sum is ioi7'3784; therefore, if 
the diameter be called feet, we muft fay there are 1017 
Square feet in this circular area, and '87S4 decimal remain¬ 
der, which, multiplied by 12, is equal to rc' 6" 5'" io iv 6 V . 
(Thus — '8784) which Sum, added to 1017 feet, (as firft 
..._ 12 ftated) the totality amounts to the en- 
“ ) ' 54 ^ tire fum of 1017 ft. 10' 6 " 5'" ro iv 6 H- 
ft*,5 The remainder, -0288, is So trivial a Sum 
——•— as not to be worth notice ; for, when it 
5 T is multiplied, it does not produce above 
•°sc-4 one-third of one in the next power. It 
—7-^-3 is upon this principle that the duode- 
-cimal (urns at the bottom of the table 
4 have been found. 
For the fake of making the table as generally ufeful as 
4 poflible. 
