1833.] Analysis of Books. 553 
of more than a figures, and consequently to be of the form eX 10” +r then if 
u® be the nearest approximate x” power to e and if ex10%—w" x10" = v and 
n.n—l 
u” x10” be supposed to expound a” and v-+r to expound na”—! b + 
2 
an— j? ...,.n, the complete nth root of ex 10” +r willbe found by finding an 
approximate nth power toe, and then seeking such a number as when substituted 
for db in nav—'! 8 + a an— 0? .... b” will render the sum of this ex- 
pression and the product of the nearest nth power already found into 10”, less or 
not greater than ex10”-+r or (a+b)”. And this operation is to be repeated 
according to the number of figures in (a+)”. 
Our books of arithmetic contain nothing farther than the above statement, and 
leave the mode of finding the second number of the root, and of its successive invo- 
lutions and multiplications into its proper co-efficients, entirely to the student. 
The Arabian arithmeticians, with a good deal of ingenuity certainly, (whether 
well or ill directed is another question,) have invented a table or diagram in 
which, by a sort of mechanical process, the sought number 4 by the bare process 
of multiplication into one figure, and addition to the number above it, is succes- 
sively involved to all its powers, multiplied into all its co-efficients, and the sum of 
the whole found. 
The Arabians give to their diagram the quaint name of Shukul-i-Mumburee, or 
Pulpit, or, as Mr. TyTLeER more grandly translates it, Anabathroidal diagram. The 
figure consists of ascending steps like those of the stairs of a Mohammadan pulpit. 
The etymologies of by far the greater part of our technical terms are not more ra- 
tional. 
The Arabian operation, in fact, is a very careful mode of finding the result of 
n. N— 
nan— 6+ Fae a™—?}.... 6% so as not to repeat any of the stepsor per- 
form the same calculation twice over. With our present improved methods, 
it is seldom that the arithmetical extraction of roots of high powers is 
performed; but were it often required, we should soon find the necessity of 
attention to this matter, and of some system in arranging our operations, so as to 
avoid doing the same thing over and over again. 
Such mechanical contrivances have been employed by the greatest Mathemati- 
cians: it will be sufficient to instance the celebrated square, almost on tke prin- 
ciples of a magic square, invented by Sir I. Newron, forsolving equations by means 
of converging series- A mind curious in tracing analogies, might discover in the 
Arabic anabathroidal diagram, some traces of that reasoning which must have led 
to the discovery of the wonderful calculating machine of Mr. BaBBaGe. 
To give an idea of the Arabian method, we shal! here extract the approximate 
6th root of 166,571,800, which is the two first steps of the example given by Mr. 
Tytrier. In the original diagram longitudinal lines are drawn between each two 
figures: for those we have substituted dots, and the several steps of the operation 
are numbered I. (which is at the bottom) II. III. &c. To abbreviate, let 10 be 
denoted by ¢, 166 by e, 571800 by 7, 2 the approximate 6th root of 166 by a and 
3 by 4, and the effect of the several operations will be as marked in the following 
diagram, 
3c 
