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THE UNIVERSITY
OF ILLINOIS
LIBRARY
|) From the collection of
» Julius Doerner, Chicago
Purchased, 1918,
SB+O-84 513.2
Gols |
MATHEMATICS UBRARY
¢'
e
1
i
fo: VV. GOODHUPF,
CHICAGO.
Entered according to Act of Congress, in the year 1885,
By 8S. W. GOODHUE,
in the Office of the Librarian of Congress, at Washington, D. C.
CHICAGO:
Printed by H. H. HOFFMANN & Co.
MATHEMATICS LIGRARL
INTRODUCTION.
1. Addition, Subtraction, Multiplication and
Division are commonly termed the four fundamental
principles of Arithmetic. Operations in Addition,
Subtraction and Division require a certain amount
of mental effort which cannot be materially dimin-
ished by any known labor-saving appliances. The
expert arithmetician may shorten the time, but he
cannot escape the labor incident to these processes.
2. But Multiplication is an abbreviated form of
Addition, the distinction being that by one process
we laboriously ascertain the sum of several unlike
numbers, while by the other we more easily compute
the sum of many like numbers. As in some
branches of industry, one man aided by machinery
may accomplish the work of 10 men, or may abridge
the time, as well as improve the quality of his own
labor in like proportion; so in Multiplication, a
skillful use of mental appliances may effect a corres-
ponding economy of time and effort.
01425
4 INTRODUCTION.
3. These points may be illustrated by two
examples; the first showing the time-saving value ~
of expertness; the second, the labor-saving efficacy
of abbreviation.
I. A tailor sews upon 13 garments, the
number of buttons expressed by the mar-
ginal figures:
How many buttons in all ?
Solomon Slowbones points with his finger
and drawls with his mouth, thus: “s-e-v-e-n
a-n-d o-n-e a-r-e e-i-g-h-t, a-n-d f-o-u-r
a-r-e t-w-e-l-v-e,” etc., while Ed. Expert,
BD DO RH KR DDR BH BH
—
Hm Co bo bo ® WW Ol WwW > bo
ee eee ae ee TU
pi
~]
holds his tongue, groups several figures
mentally, and thinks nimbly, 12, 22, 38, 48, ete.,
or taking both columns at once, 38, 78, 112, 148,
171, 208.
II. A tailor sews upon 13 coats, 16 buttons
each: How many buttons in all? This, as a
problem in Addition, requires the writing of the
number 16 thirteen times; but by the abbreviated
process of Multiplication as commonly practiced, 16
is written once with 13 under it—4 figures instead
of 26—and then brain and hand work together, as
shown on the following page.
INTRODUCTION. 5
1 6
1
Brain work 3x6=18 Hand work 8
exe to 3+1—4
£6 6 6
ag Poe | 1
8 8
6+4=10 0
I+1= 2 2
Ed. Expert, abbreviating the abbreviation, with
less brain work better directed and with no hand
work at all, thinks, as the lightning flashes,
. 19 tens and 6 threes, 208.
Has he multiplied 16 by 13? By no means.
He is exempted from so hard a task by his famil-
iarity with the relations of numbers. He has simply
apprehended 1910 as a relative approximation to
16 13, and 63 as the difference to be added.
4. With almost equal facility, in each case
substituting for a difficult computation,
AN EASY COMPUTATION AND A DIFFERENCE,
he solves the following problems:
6 INTRODUCTION.
1. 17x19 8x 40 +3 323
R27 x27 17x 40 +49 729
3. 29x84 50x50 —64 2436
1 §382x86 32x 100 —8xX56 2752
" 13286 ¥% x 8600(2867) —115 2752
5. 36X38 40x40+-8 = 9407 ee
6. 44x48 1546x100 oe 2113
7. 57X64 60X61 i 3648
8. 64x64 35+4x100 +196 4096
9. 73x89 62 x 100 +297 6497
10. 263X278 270270 4-214) 7S
5. For all factors under 100, and for multi-
tudes of greater ones, the
GEOMETRICAL BIRD METHOD
of multiplication, as developed in this work, fur-
nishes the average accountant, if not the average
school-boy, a system of simple and intelligible data
for various mental formulas, easy, rapid and accu-
rate; an insight into which will release him at once
and forever from the slow, laborious and often in-
correct drudgery of the ordinary written process.
6. By the geometrical diagrams with which the
work is illustrated, integrity of process, and accu-
INTRODUCTION. ~ 7
racy of result, are infallibly demonstrated,—the
mind being, as it were, electrified by a broad scien-
tific comprehension of the nature and scope of the
problem, as a wHoxz, instead of being confused and
lost in the multiplicity of its parts.
7. For example, with so simple a problem as
13X16, the ordinary cipherer is seldom quite sure
of his correctness. His computation has many
parts, as a tub has many staves. The tub, as a
whole, is an incomprehensibility. He can only
apprehend one staveatatime. So, fearing a hidden
leak, he carefully examines stave by stave, and joint
by joint, and finds only a negative assurance of
soundness in the lack of a positive discovery of
defect. He has no comprehensive idea of what the
product of 13X16 ought to be.
8. Let him now inspect this 3
diagram, representing the space
occupied by 13 rows of 16 but- 19
tons each, separated by crossed _
threads into 4 parts, viz:
10 tens
3 tens + and 6 threes.
6 tens
8 INTRODUCTION.
Without ciphering, he may now perceive just
what the product of 13x16 must be. His tub is
glass, and through its transparency gleams its
infallibility.
9. The general divisions of the subject are as
follows :—
I. The Multiplication Table, considered, 1st,
in its superficial aspect, as a collection of 144 simple
products to be memorized in childhood, and 2nd,
in its deeper significance, as an agency in the devel-
opment of countless larger products, by virtue of
the geometrical position, or the mechanical manipu-
lation of the figures of which it is composed.
II. Multiplication, as a language, its letters,
syllables and product-words. Slow and laborious
writing and spelling backwards, carrying and blunder-
ing, compared with quick, easy and demonstrably
accurate reading forwards.
Il. 99x99, or mental combination of several
memorized partial products as syllables of a product-
word—symbolized by the body, head and wings of
of the Geometrical Bird.
IV. Written Multiplication, or manual combi-
nations of partial products, intelligibly arranged in
INTRODUCTION. 9
the Diamond or Pyramid form. In this system,
partial products being written entire, there is
NOTHING TO CARRY,
(except in the final addition of partial products),
and consequently, a most prolific source of confu-
sion, and error is effectually avoided.
10 THE MULTIPLICATION TABLE.
THE MULTIPLICATION TABLE.
10. A small child is taught to say, and to re-
member the saying,
3 times 5 are 15.
A parrot learns the same lesson, and then brute and
human are equal. Hach repeats a form of sound
without sense, of words without meaning, and one
knows as much of arithmetic as the other.
11. The child takes another lesson—an object
lesson, of fingers on three hands, or of buttons, 3
on each of five gloves; and by actual count of the
entire 15, there comes to his mind a comprehension
of the meaning of the repeated words, a demonstra-
tion of the verity of the remembered theorem.
Relations are now changed. The boy has knowl-
edge, while the bird has only learning.
12. A few years later, the boy in partnership
with a pencil, having learned the Multiplication
Table and the Rule, is enabled, by writing 14
figures and making as many minor computations,
to cipher out the product of 99x99. He could no
THE MULTIPLICATION TABLE. La
more do this without the pencil than the pencil
could do it without him. Now, the parrot has
learning, and the boy more learning, but he
lacks knowledge. If the correctness of his cipher-
ing is questioned, he dare not affirm it, for he can-
not demonstrate it; but doubtfully reviews the
work in a blind hunt for errors. He knows not
how many 99x99 ought to be.
13. He takes a larger lesson in buttons, 100
rows of 100 buttons each, arranged as a geometrical
square. One row being taken from a side and one
from an end, 99x99 buttons remain. How many
have been removed ? Not 200, for the corner but-
ton was common to both rows. Then 200 minus 1
being taken from 100 hundred, there remain 98
hundred plus 1, or 9801. No pencil is needed to
make this computation; no external proof to verify
it. Independent, transparent, self-evident, the tub
stands squarely on its own bottom. No questioning
can now shake the boy’s confidence. He knows
what 99x99 must be; and again the child with
knowledge is superior to the parrot with learning.
14. There be multitudes of homo-parrots—school-
boys, teachers, men of letters, and men of business
12 THE MULTIPLICATION TABLE.
—with abundant learning of the multiplication
table and the rule; but with scant knowledge of the
underlying principles of Multiplication: They need
to count buttons and dissect geometrical birds.
15. A child’s fingers are too few for larger
demonstrations; and buttons in quantities are not
conveniently portable—besides being noisy and
troublesome” when dropping and _.rolling on the
school-room floor—and so it was a happy thought
of old Pythagoras, as tradition affirms, the con-
struction of a geometrical table, the little squares of
which, though convincing not the finger-tips like
buttons, yet standing for buttons to the eye, may
be counted, and thus convey to the child-mind a
practical demonstration of each proposition, from
1x 1=—1 up to the grand climax 12x 12—144,
16. The beginning of knowledge, as distinet
from learning, comes only from such actual count-
ing of tangible objects representing small products.
Later lessons may be based on immaterial buttons
in imaginary rows.*
* “Seemed washing his hands with invisible soap,
In imperceptible water.”—Hood.
THE MULTIPLICATION TABLE. 13
17. Our historic world was thousands of years
old before it saw the first Multiplication Table; and
even now there exist many tribes of men who can-
not count their own fingers and toes. A man,
wondrous learned and wise above his fellows was
he, whether Pythagoras or another, who invented
the geometrical table. But imitators now need not
his full measure of wisdom. Indeed the 144 squares
of the table may be correctly filled by a person
utterly ignorant of Multiplication, or even of Addi-
tion. He has only to number consecutively the
144 quarter-inch marks on a yard-stick, and then
transfer to the first line of the table the first 12
numbers; to the second line, the first 12 which
occur at half-inch intervals, and so on to the twelfth
line, the appropriate numbers for which are found
three inches apart.
18. Can these dry bones live? Can the soul
of numbers vivity a dull geometrical. clod thus
ignorantly conceived and mechanically developed ?
Is there in this assemblage of figures any hidden
thing beyond the 144 products which childhood
cons by rote and crones in sing-song ? Investiga-
tion proves that by virtue of their geometrical posi-
14 THE MULTIPLICATION TABLE.
tion alone, these soulless characters have marvel-
lous potency in the solution of stupendous problems,
invisible if not incomprehensible to the childish
student.*
19. Here, for example, is a miniature table in
which the ordinary observer sees nothing beyond
2x2—4. If we add together the upper numbers,
1 and 2, and the side numbers 1 and 2—the number
1 doing double duty—we have 3x3, the product of
which equals the sum of all the num-
bers in the table. If, instead of add-
ing the factor figures, we consider
them as expressing the number 12,
then the product of 12x12 appears
by turning the point a to the left and adding per-
pendicularly. Ifthe point a be turned to the right
and the reversed figures be read as 21X21, the
product 441, appears by adding as before. If the
point a be turned upward or downward, and the ©
factors be read as 12X21, the product 252 is simi-
larly obtained.
20. In the next table may be found the pro-
* **There are more things in heaven and earth and the multipli-
cation table, Horatio, than are dreampt of in your philosophy.”—
Shakespeare & Co.
THE MULTIPLICATION TABLE. 15
duct of 1+2x1+2-+3, or 3x6=the sum of all
the numbers in the table; also with « at the
Left, 12x 123=—1476
Bottom, 12 321=3852
Right, 21 x 321=6741
Top, 21 x 123=2583
If the side factors 1 and 2 be taken as factors
only, and not included in the addition of products,
we have with a at the
Left, 12 23=276
Bottom, 12« 32—=384
Right, 21 x 32=672
Top, 21 « 23=483
21. In this table may be found the products of
4x 4=16
13x 13=169
13x 31=403
31x 31=961
With a cipher in each vacant square may be
developed the products of
103 x 103=10609
301 x 103=31003
301 301=90601
16 THE MULTIPLICATION TABLE.
22. In this section representing the four central
squares of the table, under and at the right of the
factors 6 and 7, the product of
6+7x6+7, or 13X13 equals 3
the sum of the four numbers, | 6
169. Addition with the point 4 a3
a turned to the left developes ) 9
the product of- 67 x 67 = 4489.
“
A change in the position of a a ‘
two figures, developes in the = E
second diagram the product of - f
607 x 607 = 368449. 2,;4 9
23. This section represents the four squares at
the right lower corner of the table, under and at
the right of the factors 11
; os Bl 10
and 12. The sum of the 1 1
four numbers 529 is the 9 3
product of 11+12x11-+4 1 -
12 or 23X23. The pro-
duct of 1112 x 1112 —
1236544 is obtained by
addition with the point a
turned to the left. The factor numbers 11
ll and 12 overlapped as in the margin 12
form by addition the number 122. 122
THE MULTIPLICATION TABLE. 17
The numbers in the “ l
second diagram similarly 9
overlapped, develop by ad-
dition, the point a being pel hee
3 4
turned to the left, the pro- |
duct of 122x 122—14884. 2
24, Finally, the whole Multiplication Table
may be regarded as a single problem, one factor
being the sum of the numbers in the upper line, the
other, of those in the left column, while the pro-
duct is the sum of the entire 144 numbers. , This
problem may seem at first sight a formidable one;
but the elements of its solution are wonderfully
simple. The upper line of the table may be seen to
consist of six pairs of numbers, viz: 112, 211, 3 10,
49,58and6 7; the sum of each pair being 13,
the average of each number 63, and the sum total
78. In the second line these proportions are
doubled, and so on by a regular progression to the
twelfth line. Conditions precisely similar appear if
the table is scanned in columns.
25. Itfollows that the square of 64 or 42+ must
bethe average number for the entire table, and there-
fore the product of 421 by 144 must be the sum
18 THE MULTIPLICATION TABLE.
total: also, that the same aggregate must result
from squaring the number 78. By methods here-
after explained, the square of 78 is mentally de-
duced from the square of 75 or of 80—an easy
computation and a difference—thus, 5634+450=
6084, or 6404—320—6084. The problem 424 x 144
divested of its fractional complication by substitut-
ing the equivalent factors 169 x 36 is solved at sight,
thus: 4854+1230—6084.
26. As the number 169, the square of 13, repre-
sents the average aggregate of four squares, the
question naturally arises, are there in the table any
four numbers of exactly that sum? Itis a curious
fact that the entire table is made up of 36 such
sroups of four—one odd number and three even
ones in each group—all symmetrically arranged
around a common center. The figures comprising
each group are equi-distant from the center of the
table, and also equi-distant from each other; and
stand at the angles of a perfect geometrical square
—unless the table itself varies from that form.
The sum of a side of one of these squares is invari-
ably a multiple of 13; and the product of the largest
by the smallest number is in every case equal to
that of the other two.
THE MULTIPLICATION TABLE. 19
These groups of 169 may be readily located by
reference to the table (28), thus:
1 12 2 24 Si Saplé 4 48
12 144 Blot 10 120 9F108
By analysis, the fourth group is seen to consist
of
4-1s and 4 12s
wersesetand 9. 1¢
13 13 13 13
MULTIPLICATION TABLE.
1
2! 4] 6] 8/10/12/14]16]18] 20] 29) 24
3] 6] g/12/15/18/21|24|27] 30) 33) 36
~4{ 8{12] 16/20 | 24/28 | 32 40; 44} 48!
610] 15 | 20 | 95| 30/35 140145] 50) 55] 60
6/12] 18 | 24| 30 | 36/42/48] 54] 60) 66] 72
714] 21 | 28 |35 | 42] 49 | 56 | 63 | 70] 77] 84
8 {16 | 24/32 | 40 | 48 | 56 | 64] 72] 80] 88] 96
ee | | | | | es | |
io)
—
So
bo
we
(SX)
3)
eS
es)
pha
ce
On
(=P)
(one)
as
90 |{0Q | 110/120
re)
12 | 24 | 36 | 48 108
120] 132/144
20 THE MULTIPLICATION TABLE.
27. The same principle prevails in any square
multiplication table, whatever its magnitude, the
sum of the group of four numbers being in any case
equal to the next square number outside the table.
In an odd-numbered table, as 5X5 or 9X9, the
central figure, 9 or 25, represents the average num-
ber, and is surrounded, the first by 6 groups of 36
each, the second by 20 groups of 100 each.
28. In the accompanying table of 12x12
squares, heavy lines at 9x9 indicate the extent of a
table of unit factors. Factors too large to be multi-
plied mentally are necessarily subjected to a written
process, by which several partial products are
obtained and added together. For the numbers
representing these partial products, the operator
depends almost invariably, not on original compu-
tations but on the remembered lessons of this
lesser table.
29. We may imagine a person unable, from
some mental defect, to memorize these lessons and
yet able by using this table for reference to obtain
and set in order for addition the partial products of
factors of any magnitude by means of an improvised
geometrical table like the one here represented. In
THE MULTIPLICATION TABLE. 21
this table the product
of 684 x 397 is obtained
by placing one factor
at the top, one at the
left, the 9 partial pro-
ducts within the 9
squares, and their sum
971,548, extended
around the bottom and right of the table, the point
a being uppermost during the addition.
30. By the ordinary process, instead of 9, then
are developed 3 partial products—which may be
traced in the 3 lines of squares in this table, viz:
7X 684 4788
90 x 684 6156
300 x 684 2052
271548
31. By either method, only the significant.
figures of the partial products are written, the final
ciphers being omitted. The full expression of the
partial products in the table requires that the
ciphers in each square be placed at the right of the
significant figures.
22 THE MULTIPLICATION TABLE.
Thus there appear:
1 product of units or ones, 28
Dae Po tens 56, 36, 92
ote hundreds, 42, 72,* 12 126
EAS ry ten hundreds, 54, 24 . 78
iss? hundred-hundreds 18
271548
32. An observable feature in the Multiplication
Table (28), is the diagonal line of square numbers,
1, 4, 9, 16, etc. A square number being the pro-
duct of equal factors, has but one place in the table,
but other products are duplicated by the inversion
of their factors, the No. 132, for example, appearing
both as the product of 11x12 and of 12x11.
Hence the two sections of the table as divided by
this line, are exact duplicates. From a table minus
one section, a boy minus one eye might learn the
144 products; but both sections as well as both .
eyes are usually considered desirable.
33. Diagonal lines parallel to that of square
numbers, consist of numbers diminished by square
numbers. Thus in
5 lines opposite 36 are 35 32 27 20 I1
the differ’ce from 36 being 1 4 Ee
* 72is a product of ten-tens, identical with hundreds.
THE MULTIPLICATION TABLE. 23
- THE TABLE OF SQUARE NUMBERS.
34. A method of Multiplication without multi-
plying is based on the use of a reference table of
square numbers, written or printed, combined with
a process of Addition, Subtraction, and Division.
35. Consecutive square numbers are identical
with the sums of consecutive odd numbers. Hence
a person ignorant of Multiplication may construct
this table by a series of additions, thus:
Numbers, | yi 5 4. 5 6
Squares, 1+3—4+5—9+7=—16+9=25+11=36
As the work advances, its correctness may be
tested at every 10th number. Thus 4x4 being 16,
40x40 must be 1600. To change 40x40 buttons
to 4141 buttons, 2 sides and a corner must be
covered or 2X40+1, making the square of 41=—
1681. Then to change the square of 41 to that of
42, the next odd number 83, must be added making
1764.
36. Square numbers, except the number 1, are
either even, and multiples of 4, or odd, multiples of
4 plus 1. The first 10 square numbers will be
found on the following page.
24 THE TABLE OF SQUARE NUMBERS.
1 4,..9 16 25.36 49° G25 sie
The terminal figures of these 10 are repeated in
every succeeding 10. Thus if a number ends with
0) its square ends with 00
00 0000
1 or 9 1 remove from 0 1
2 or 8 2 * 4
DOr. 3 . 9
4 or 6 4 a 6
5 5 A 25
No square number ends with 2, 8, 3 or 7, or with
an odd number of ciphers, or with 5 other than 25.
These characteristics may be verified by the accom-
panying table, which contains the squares of all
numbers up to 99.
25
THE TABLE OF SQUARE NUMBERS.
TABLE OF SQUARE NUMBERS,
F8IG | LPOG
VGLO | 19E9
VHSE | LéELE
FOLGE | 1096
POLL | T89T
FCOL | 196
V8P LFF
VFI LéT
iy {
G I
v9FS | 18Eé8
_—————
O0OF9
OO6F
0096
0066
0091
0018 06
08
OL
09
OG
OF
0€
26 THE TABLE OF SQUARE NUMBERS.
37. The method of obtaining products by refer-
ence to this table is based on the fact that the
square of the sum of 2 factors includes the square
of their difference, plus 4 times their product.
The process is as follows:
I. Find the square of the sum of the factors.
II. Find the square of their difference.
111. Subtract one from the other.
IV. Divide the remainder by 4
Example I. 9x3.
9+3=12 Square of 12, 144
9—3=—6 Square of 6, 36
Remainder, 108
4 of remainder, 27
38. Demonstration. If 4 cards of buttons,
3x9 on each card, be
placed together as repre- | a ae
sented in the accompany-
ing diagram, they form, 6
1. An outer square,
measured on each side by
12 buttons.
2. An inner vacant
square, measured on each side by 6 buttons.
THE TABLE OF QUARTER-SQUARES. 27
Example I. 57 x 38.
57+38—95. Square of 95, 9025
57—38=19. Square of 19, 361
Remainder, 8664
3 of remainder, 2166
TABLE OF QUARTER-SQUARES.
39. The necessity for a final division by 4, in
each operation, has been obviated by dividing the
entire table by 4, or in other words, constructing a
table of quarter-squares.
40. The quarter-square of any number is equal
to the square of half that number. Thus the
quarter square of 12 is the square of 6, or 36; of
13, the square of 64 or 424. In this system, how-
ever, the fraction + pertaining to the quarter-squares
of odd numbers is dropped without affecting the
result. The proximate quarter-square is the pro-
duct of the two numbers proximate to the half, as
of 13, 6X 7=42.
41. This table, like the table of square num-
bers, may be constructed by successive additions ;
the consecutive quarter-squares of even numbers
being identical with the sums of consecutive odd
numbers, and vice versa, per example on next page.
28 THE TABLE OF QUARTER-SQUARES.
ven Nos. , 4 Bose 10 ‘12
4 Squares, 143= 4+5=9+7=1649=25+11=36
Odd Nos, 3 5 i 9 11 13
a Syrs., 2+4—6+6=12+8=20+10=—30+12=42
42. By reference to a table of quarter-squares,
products are obtained as follows:
x, 1. 9X 3.
9+3=12. Quarter-square, 36
9—3—6. 7 9
Remainder, 27
Ex. IT. 57 X 38.
57+38=—95. Quarter-square, 2256
57—38=—19. 2 90
Remainder, 2166
43. This method has the following earnest com-
mendation, in “Sane’s Hicguer Aritumetic.” Kd.
and Lond, 1857:
“M. Anton Voisin published in 1817 a table of
quarter-squares of all numbers up to 20000, which
enables us to find the product of any two numbers
up to 1000. Mr. Laundy has lately published a
table of quarter-squares up to 100000. This work
should be in the hands of every professional cal-
culator.”’
Popular judgment, however, has consigned the
system to oblivion, from which it is here resurrected
in compliment to its ingeniousness rather than to
its usefulness.
MULTIPLICATION AS A LANGUAGE. 29
MULTIPLICATION AS A LANGUAGE.
44, Multiplication may be considered as a lan-
suage, whereof the numerals are letters; the par-
tial products, syllables, and the sum of the sylla-
bles, product-words.
45. The alphabet of this language consists of
the 10 numerals,
0 1 2 3 + 5 6 7 8 4
commonly termed the cipher and the 9 digits. The
digits are also called significant figures.
46. A syllable is the product of two numbers,
as 9X9, or any lesser factors. No larger syllables
than those contained in the lesser Multiplication
table, (28) occur in the ordinary written process ;
but in mental computation, any product may be
treated as a syllable when recalled by an act of
memory, or when instantly computable, as of 12x
12, 15x18, 25x 25, 75 x 64.
47. Letters or syllables followed by ciphers
assume magnified proportions, but are not thereby
rendered more difficult of computation, Thus in the
30 SPELLING BACKWARDS.
formula 80x 1200=96000, the 3 ciphers from the 2
factors mechanically supplement the simple product
of 8x12, just as 3 dots are made to complete the
writing of the word Division.
48. In any product word the number of sylla-
bles equals the product of the number of digits in
one factor by the number in the other, thus: 2, 3,
or 4 digits in each factor produce respectively 4, 9,
or 16 syllables; while 2, 3, or 4 digits multiplied
by 1 digit produce 2, 3, or 4 syllables.
es ae
SPELLING BACKWARDS.
49. The word ‘‘ Multiplication”’ requires for
its writing, some 50 distinct movements of pen or
pencil, for its spelling, 14 letters; for its pro-
nouncing, 5 syllables; but for its silent reading,
only the passing glance of a practiced eye.
50. The various methods of Multiplication, as
applied to factors under 100; for example, 46 x 48,
exhibit contrasts equally striking. The ordinary
computer, ignorant of syllables, takes the slowest
CARRYING AND BLUNDERING. 31
and most toilsonie method, wasting time and
energy on 50 manual movements in 14 figures and
2 lines, meanwhile making 10 computations—thus
laboriously writing backwards with many letters a
word which may be instantly read forwards with
2 syllables, thus 44x50+4x2. When all is done
it is only half done. There are 24 chances of error
in his 14 figures and 10 computations, and he
seldom ventures to omit reviewing the entire work
in search of the possible mistake.
ep eres iSeeate
CARRYING AND BLUNDERING.
51. The written process prescribed in the “Rule’’
of the school-books, although theoretically sound
and philosophically correct, is practically the occas-
sion of infinite blundering. While sufficiently
simple to be apprehended and practiced by children
7 years of age, it is yet so complex as hardly to be
comprehended by the same children when that age
is doubled.
“ Likewise ye Monkie, albeit he turneth about
y. Crank of ye Hande-Organ, hath natheless Small
Knowledge of ye Musickal Product thereof.’’—
Antique Author.
32 CARRYING AND BLUNDERING.
52. The hidden spring of this defect is the
ignoring and the unconscious mutilation of the
natural syllables of a product. A child, unable to
read the word, Multiplication, may yet succeed with
- the syllables, Mul-ti-pli-ca-tion, and again be sorely
puzzled by the no-syllables, Mu-lt-ip-li-cat-ion. So
a child may comprehend the 9 syllables of the fac-
tors 684 « 397, either within or without the lines of
a geometrical table (29, 117); and may further
apprehend the fact that each syllable is unques-
tionably right and in its right place; and thata
correct addition only is required to insure a correct
product. |
53. Turning from these 9 simple syllables, all
verified by the multiplication table, to the three
complex partial products (30); each consisting of
syllables butchered as soon as born—the tens
chopped off from one and stuck on to another—what
can the child make of them? They may be right,
if the ciphering is right ; as the tune may be ground
out if the crank is properly turned; but neither child
nor monkey can distinguish the right from the
wrong.
54. The school-boy, by the time he reaches the
‘‘first class,” is so thoroughly satwrated with the
CARRYING AND BLUNDERING. 33
Addition and Multiplication tables, that he cannot
miss the resulting syllable of 8-+-8 or 4x4. He
fails, not in setting down the 6, but in carrying the 1.
55. The device of carryine, the child’s riddle,
and the man’s life-long stumbling-block, is the
source of nine-tenths of the every-day blundering
in Multiplication.
56. If the world were restricted to one tool, it
should be the jack-knife, which fits the universal
pocket of boy or man, and is pre-eminently the
implement of all work; if to one method of Multi-
plication, the old one, with its little and big blades,
the ‘‘table’’ and the ‘‘ rule,”’ has been fitted to the
universal head, and like the ox-team, is to be com-
mended for the slow and heavy labor it has accom-
plished in the past.
57. But, the jack-knife being too delicate for
the ship-carpenter and too clumsy for the surgeon,
the broad-ax and the lancet have been evolved; and
the ox-express is displaced by steam and electricity.
So for the practical computer the Multiplication
table is too short, the old written process too long
and laborious. The child’s table, 12x12, should
be supplemented by the man’s table, 99x99, not
34 LIGHTNING CALCULATORS.
written, not memorized, but instantly computable;
and the written process—never applied to factors
under !00--shortened, simplified and freed: from
the errors incident to carrying, should partake more
of the electric alertness of the brain, less of the
dull plodding of the hand.
LIGHTNING CALCULATORS.
58. Partly in this direction, and partly oppo-
site thereto, is the method of the ‘lightning caleu-
lator’’; combining on the one hand more head-
work with less hand-work, and on the other greater
speed with less simplicity, By this method the
product is written directly in one line, without the
intervention of partial products.
Ex. 684 x 397. 684
397
Written, 271548
Performed—the numbers carried being expressed
in Italics : —
7x4 28
7X8 and 9x4 +2 94
7x6 and9x8and3x4 +49 135
9x6 and 3x8 +13 91
3X6 +9 27
LIGHTNING CALCULATORS. 35
59. The student may be assisted in compre-
hending this process by practicing the following ex-
amples,in which the digits correspond to the several
orders of units numbered from right to left.
i Dees 321 X 321.
Ex. IT. 4321 x 4321.
Ex. WI. 54321 54321.
Ex. I is written, _ 321
321
103041
Performed:
1x1 1 (1 product of units.
1x2 and 2x1. 4 (2 products of tens.
1X3 and 2X2 and 3X1 IO (3 “ hundreds.
2x3 and 3x2 +] 13 (a? thousds.
brag +i 10 (1 “ 10 thousds,
60. This operation may be made partly mechan-
ical, by writing one of the factors with the digits
transposed, thus: 123 on a moveable slip of
paper and passing it from right to left under. the
other factor, taking the products, or sums of pro-
ducts, of digits perpendicular to each other in each
of 5 positions, as follows:
Position : 5 4. 3 2 1
Nature of Products. 1o-thousands. thousands. hundreds. tens. units.
321 321 321 A Ee WA |
123 123 123 123 123
ee ee
mews arc (10.4
36 LIGHTNING CALCULATORS.
61. Arithmeticians generally are aware of the
existence of this method; but comparatively few
conquer the first discouragements attending its
practice. Lightning calculators, like poets, are
born, not made; and their exceptional powers can-
not be transferred to their less gifted admirers.
With the operator of only average capacity, this
method increases the lability to error, and dimin-
ishes the facility of its detection.
‘¢And yet shew I unto you a more excellent
way.’—I Cor. XII, 31.
THE GEOMETRICAL BIRD. 37
THE GEOMETRICAL BIRD.
ee
62. In the miniature Multiplication Table (19),
the product of 1212 is developed by adding the
four numbers in a certain direction.
The accompanying diagram instead of vias
consisting of four squares of uniform boy 4
size, represents the space occupied by
12 rows of 12 buttons each, 10 2
separated by crossed threads
into four sections, correspond-
ing to arithmetical syllables.
The relative proportion of these
syllables suggests the similitude
of four parts of a bird, viz:
1 ten-ten, the body, 100 buttons.
2 tens, a wing, PAB ied
2 tens, a wing, open
2 twos, the head, Ao
63. Dispensing with the formal lines of a table,
a ready method is here suggested of obtaining the
products of factors from 11 to 19, by either a
written or a mental process, as shown at the top of
next page:
38 THE GEOMETRICAL BIRD.
Written process: Mental process:
1
EEX 13S salero 11+3 tens, 143
3 1x3 units
143
2
ISSAG AAG ae 13+6 tens, 208
6 3x6 units,
208
7
1 Xa 17+7 tens, 989
ff 7X7 units,
289
5
15x19 1456 15+9 tens, 28h
9 5x9 units, ;
285
64, The same method may be applied to larger —
factors, but not with any practical advantage, thus:
4
4 4
4 4 3) 4
24 34 3 44
24X27 28 34x37 328 34x47 3h or 4.28
BY 37 3 47
7 7 7
er 7 7
648
1258 1598
THE GEOMETRICAL BIRD. 39
65. ae
es 400 x 400 NM
>
=
400 40 4
‘¢ Hach one had six wings; with twain he covered
his face, and with twain he covered his feet, and
with twain he did fly.” —Isa. VI, 2-3.
Diagrams on the same scale representing the squares
of 4444 and of 44444 would measure on each side,
the first 22 inches. the second, 18 feet.
119. Product syllables arrranged for addition
must each be represented by 2 figures, a cipher
70 THE PYRAMID FORM.
being prefixed when the syllable is less than 10,
except that at the extreme left the cipher may be
omitted.
Example 31632. a Es
3X3 09
SC Os ulin oO 1803
DS OKy LO, Oe 020618
32 12 62 32 9013609
1X3, 6561)3 506 030618
6x3, 3X1 1803
3X3 09
10004569
120.. The number of figures in any product is
either equal to the number in both factors, asin the
squares of 32, 317 and 3163, or one figure less, as
in the squares of 31, 316 and 3162.
THE PYRAMID FORM.
121. The diamond form may be condensed, and
the partial products arranged in pyramid form, by
coupling syllables of like order.
THE PYRAMID FORM. Le
Example 684 x 397. 6 8 4
397
6xX7+3x4 54
6x9+3x8, 8x7+9x4 7892
6x3 8x9 4X7 187228
271548
Coupled syllables may be written separately,
except at the extreme left, when their sum exceeds
99, as in the following example:
Ex. 897X789. 789
897
7X7+8x9 121
7X9+8Xx 8534 12739
meres; 9X7 567263
707733
122. In squaring a number it should be remem-
bered that one factor must be doubled in computing
wing products.
Ex. 438. 4 38
8x8 64
3x3, 6X8 2448
42 32 So 160964
191844
THE INVERTED PYRAMID FORM.
THE INVERTED PYRAMID FORM.
123. Practically, this form is preferable to the
preceding ones, as the products of digits in perpen-
dicular lines are taken first, and thereby the posi-
tions of the several orders of units are determined
at the outset.
Ex. I 721834 Yaa a
8 3 4
7x8 2x3 Tose 4 560604
7X38 2) 94 35] 3711
7x4+8x1 36
601314
Ex. Il. 4372 8694. 8694
; 4372
8x4 6x3 9x7 4X2 32186308
8x 3+4x6, 6xX713x4, 9Xx217x4 486946
8x714x9, 6X2+43x«4 92294
8x2+4x«4 32
38010168
124. As each digit in a factor is represented
by 2 figures in its product syllable, so each cipher
must be represented by 2 ciphers.
THE INVERTED PYRAMID FORM. 73
Hx. I. 2703x8976. 2703
Ex. IT. 3024 2850. S-¥eieG
302 4 . 12
28 5 0 1442
— 184900
06001000 16630018
241620 560021
1932 0027
08 24
8618400 24262128
125. In examples lke the following, the entire
product may be written in one line, or may be com-
puted mentally.
Ex. 307 x 906 7009 x 4008
307 i 0;,.059
9 0 6 4008
278142 28092072
126. With factors of an unequal number of
digits the product forms are incomplete.
Ex. 6789 x 24. 6789
24
6789 24
24 1228
$s * 1432
01636 * 1636
* 1450 ke
1228 es
24 *
162936 162936
74 RAPID MULTIPLICATION.
RAPID MULTIPLICATION
127. Expert mental computers may abridge
the written formula, and perform more rapid work
by dividing the factors partially or wholly into sec-
tions or periods of 2 figures each, thus making the
product syllables larger in amount and less in
number. The product of a period of 2 figures must
occupy 4 places, ciphers being prefixed if occasion
requires.
Ex. 1312x914. Lge
ooied
139, 12X14 1170168
13x 14+12x9 (”) 290
1199168
269 X 248. 269
248
26x24, 9x8 62472
25x17—1 ~~ (°) 424
66712
45672, 4567
452 672 20254489
90X67 (°°) §030
20857489
MULTIPLICATION. BY SUBTRACTION. 75
7128x7246. peed) ities
7-9 486
“x7, 12x24, 8x6, 49028848
area (2) 12% 22 (°) 952264
Tx 14-(°°) 98
51649488
MULTIPLICATION BY SUBTRACTION.
128. The product of any number multiplied by
nines may be determined by annexing a cipher for
each nine and then subtracting the original
number.
Ex. I. 76499. 76400
764
75636
DemonstTraTIon. 764 ones from 764 hundreds
leave 764 ninety-nines.
ieee e900 C099,
9999000 or 9990000
9999 wou
9989001 9989001
76 MULTIPLICATION BY SUBTRACTION.
129. With factors of a like number of figures,
the process—either written or mental—may be
abridged by annexing to a number J less than the
multiplicand the successive remainders obtained by
subtracting each figure of that number from nine,
commencing at the left.
Examples: |
68x99 6732
695 999 694805
7890 x 9999 78892110
7891x9999 78902109
The first example is thus performed:
67 with (6 from 9) 3 and (7 from 9) 2 annexed.
130. Demonstration. 68 rows of 99 buttons each,
changed by taking one row from the side and adding
one to the end became 67 rows of 100 each. The
side row, 99, being 32 in excess of the end row, 67,
the total number must be 6732.
Again, the inner factors, 68 x 99,
exceed the outer factors, 67: °>e aie
by the product of differences (from 68)l1 xX 32.
See Relative Factors.
BREVITY. ak
lA ae
131. Philological “Josh Billings” playfully pities
punctilious pundits, who with infinite labor acquire
just education enough to spell a word in one partic-
ular way; but not enough to indulge themselves in
66
that variety of forms wherewith he, “ good easy
man,’ may give his word a new dress—usually a
very short one—for every day in the week.
132. The adept in the various short and easy
methods developed in this work, may well be moved
to a more hearty compassion for the unprogressive
cipherer who knows only one process of multiplica-
tion, and that a laborious, blind, and often inaccur-
ate one. Whether ‘‘fonetik orthografy’”’ shall ever
overcome popular opposition or not, no reasonable
objection can be made to shorter, surer, and more
intelligble methods of spelling products. The
brevity of free and easy spelling is illustrated in the
treatment of the following 10 problems, each of
which is solved in 2 or more different ways. The .
operations extend only to the development of partial
products from which the full products may usually
be read from left to right.
78 BREVITY.
Problem. First Process. Second Process.
l. 1962. 36186 40016
228 1s
LR TO OE: 17412 1) 218
526 40% 872
3. 825x426. 320650 (8) 3550
308 (—19%) 3550
4. 9864297. (25) C50). pn
| at x28} 76 gov 5728
ee x64 1728
De SLIT Sas 13606 « 103 17716
67 —>x 20 344
6) 1400 aoe 13524 x 40 18120
| 369 5% . 906
26428 X77); 4482100 x750 4821000
4996 < 200eteu
8. 46.52 1642.25 1 2325 a.
520 =e 162.75
9. . 32502. 9062500 4 10833333* B.
150 — le ee
1GS6 C87 Be 121645625 fc 1484375 9c
176 —59% 7421875
1200 :
1650
} Third Process.
A. 46X47 and .25 2162.25
B. 3233 and 2500 10562500
Cc. 118-+19.5 and 752 137505625
18 19.5 351
* This product must be expressed by 8 figures, the last 4 being
2500 (119, 36.)
ERRORS. 79
ERRORS:
133. The way to avoid errors in computation is
_to avoid them.* There is no other way. ‘The
"process of multiplication being learned in childhood,
if in after life all its conditions be strictly observed,
error is impossible, and each and every attempted pro-
duct must infallibly be right. Failing this, the
defect is not in knowledge nor in memory, but in
attention and precision. When Grandame ‘‘ drops
a stitch,’ it is not because knitting is too intricate
or too laborious. Nothing can be easier than for
her to make a correct stitch; but in the monotonous
making of a thousand, she becomes tired or sleepy
or inattentive. Her boys and girls all inherit in a
greater or lesser degree, this trick of inattention;
and consequently as errors in computation are
likely to continue for some time to come, methods
must be practiced for their detection and correction.
_ * Horace Greely said: “The way to resume specie payments,
is to resume.”
80 PROOFS WHICH DO NOT PROVE.
PROOFS WHICH DO NOT PROVE,
134. Under the fallacious title of PROOFS, some
arithmeticians have prescribed certain forms, which
like the forms of criminal jurisprudence, sometimes
result ina conviction of the wrong, never in a vindi-
cation of the right. When a prisoner is not proven
to be guilty, his innocence may be legally assumed,
although that also is unproven, and so when acom-
putation is not convicted of error, the evidence of
its correctness is only negative.
135. These methods of ‘‘ proof’’ are
30.
1. Inverting the factors.
2. Dividing the product by a factor.
3. Casting out the nines.
By inverting the factors the multiplier in the first
operation becomes the multiplicand in the second;
new partial products are introduced, and the agree-
ment of the two products establishes a preswmption
of correctness, strong or weak, according to the
proficiency of the operator.
136. Dividing the product by one factor gives
the other factor as a quotient; provided, that both
PROOFS WHICH DO NOT PROVE. 81
operations are correct, which is not proven, and
also provided that if both are incorrect, one error
counterbalances another.
137. Ben. Blunderhead tries his hand at these
two methods of proof, with the following result—
his ciphering being omitted :
Examples: Proof.
78 X 43=3344 43 x 78=3254
68 x 92—6246 6246+68= 99
As the two products of the first example do not
agree, there must be an error either in the compu-
tation or the proof, or both. He has doubled his work
to no good purpose, and now his only remedy is to
review carefully what he has performed carelessly.
In the second example he has written 14 figures to
obtain a false product, and 17 more to prove it by a
false quotient. Ben’s chief recommendation for a
clerkship is the fact that he can prove, in the last
half of each day, the computations made in the first
half.
82 CASTING OUT THE NINES.
CASTING OUT THE NINES.
138. The question, how the correctness of a
computation could be demonstrated by casting out
the nines, has been to the partially informed, a
mystery deep and awful as that of witchcraft, or of
the casting out of devils. When the demonstration
proves to be no demonstration, then the mystery
vanishes.
139. Hvery ten equals 9 and 1; every hundred,
11 nines and 1; every thousand, 111 nines and 1.
Hence the number, 7468 contains
In 7000, 777 nines and 7,
In 400, 44 nines and 4,
In 60, 6 nines and 6,
In 8, é 8.
The 4 remainders are duplicates of the original digits,
and in their sum, 25, are 2 additional nines and a
final remainder of 7. This final remainder is the
essential object of search, the number of nines being
immaterial; and as the digits of any number may
CASTING OUT THE NINES. 83
be considered as so many remainders in excess of
an unknown number of nines, it is easier to cast the
nines from them than to divide the whole number
_by 9. The casting consists in dropping 9 at every
opportunity from the sum of the digits, as in the
number 7468, 7+4=—2+6—8 x8=7, or taking 8,
6, and 4 as 2 nines, the number 7 stands alone as a
remainder.
140. An operation in multiplication is tested by
casting the nines from each factor, multiplying the
remainders together and casting the nines from their
product. Then casting the nines from the main
product, the two remainders should be equal.
Their inequality demonstrates the existence of
error. Then equality demonstrates—nothing! The
product may be right: it may be wrong: but
according to the theory of the test, it is presumably
correct.
141. This device serves as a detective for errors
not measurable by the number 9; but any falsity,
however glaring, which involves a multiple of 9—
including all transpositions of figures—may pass
the test unchallenged, as appears by the example
shown on the following page.
84 CASTING OUT THE NINES.
B56 was.
423 4
32122728
403657
6038
60
36823098
1 Error, 36822098
2 Errors, 37822098
Tranposition, 36832098
9 instead of 0, 36823998
0 omitted, 3682398
Assumed product, 12
Remainder 3
6é
4
12 Remamder, 3
Remainder 3, Presumably correct.
2, Error demonstrated.
3, Presumably correct.
If we know that 12 is not the true product, we
know it by insight. Itis not disproved by casting
out the nines. Language should not be perverted
by applying the term proof to such a piece of no-
evidence.
DEMONSTRATION. 85
DEMONSTRATION,
142. The positive evidence that an assumed
product is correct, as distinguished from the nega-
tive evidence that no error is detected, has its primi-
tive foundations in the actual counting of tan-
gible objects representing small products. Every
civilized child sooner or later demonstrates by his
fingers that 2 fives are ten. The Multiplication
Table is a systematic collection of small products
which have been demonstrated in the counting of
their squares by thousands of children. Memor-
izing this table in childhood obviates the necessity
of innumerable demonstrations by actual counting,
otherwise required in the daily affairs of life.
143. From the known in the Multiplication
Table, the child may push his demonstrations out-
ward to the unknown, thus: If 12 times 9 are 108,
36 times nine are 3 hundreds and 3 eights; or if 12
times 12 are 144, 24 times 12 must be 288. The
demonstrations of the problems
13 x 16=208 ( 8)
99 x 99=9801(13)
86 DEMONSTRATION.
are brought within the mental grasp of childhood.
In fact, the Geometrical Bird method, for all num-
bers under 100, is from first to last to the adult
mind, if not to the child mind, a self-demonstrating
method.
144. We have witnessed the unsuccessful
wrestling of Ben. Blunderhead with the problem,
68x92. (137) Tid. Expert solves it thus:
80% 8012513
or (0X) 902% 22-) O26
or 60X 100+ 8x32
His mind grasps the problem as a whole: the
operation is self-demonstrating; and he knows
that error therein is impossible. But Ben. fails to
comprehend the process, and so to him there is no
demonstration; yet there is even for him a partial
demonstration. The product is the sum of the
accompanying 4 partial products, which his memory
may draw from the multiplication table.
Thus far, he may comprehend that the 72
work is correct beyond a peradventure. So 54/6
renouncing all ‘‘ ways that are dark, and 12
9
tricks that are vain,’ such as mutilating
partial products, and carrying from one to another,
DEMONSTRATION. 87
or inverting the factors, or dividing the product, or
casting out the nines, if he can now trust himself
to read the product-word from the product syllables,
or to add these numbers correctly, his task will be
accomplished, the only element of uncertainty being
in this addition.
145. To larger minds than Ben’s, larger prob-
lems, as 684 x 397 may not be demonstrable as a
whole, but the nine partial products (117), or the
6 partial products (121), may be examined and
cross-examined one by one from right to left, or from
top to bottom, and thus verified as absolutely right.
Then if correctly added, their sum cannot be other
than the true product.
146. Notso with the 3 partial products (30),
obtained by mutilation of syllables and carrying.
They cannot besoimplicitly trusted. They are right
if they are not wrong, but begging the question is
not sound logic. Universal experience shows that
this method is everywhere and always a fruitful
source of error.
MULTIPLICATION.
88
CO ri C2
Ominrand
DW l= Ors NM OO HH
DrOlowiNa aA {Yen}
DWP O1O HNO ww 10 SH eS
DOI TNO SH <1 69 6 faa
Drowdrsio sH OD 1IDSH CO RAM
Dr OnmNwolo H om 1d CO OMm
DMP OI AND SH SH 10 69
TD 6 10 41 GR OD SH ID Heo
Dre Oily iNow cH 1
Drornc sas
Ot IANO
A TRANSPARENT PROBLEM.
APPENDIX, 89
APPENDIX.
The Combination Table.
A case has been supposed (29), of a person
unable, from some mental defect, to memorize the
multiplication table, and yet able by reference
thereto to obtain and set in order for addition the
partial products of factors of any magnitude, by
means of an improvised geometrical table.
Proceeding on this principle, a combination of
unit tables may be so arrangedas to indicate by means
purely mechanical, the partial products of any and
all factors, leaving the operator only the task of
adding them. |
The accompanying table of four sections,
one representing units,
two tens,
one hundreds,
consists of unit tables so inverted and arranged that
factor digits, in distinctive type, run on internal
lines, one horizontal and the other perpendicular,
in order that any required partial products may be
inclosed within a hollow square or rectangle. This
table is sufficient for all factors up to 99. Factors
of 3, 4, or 5 figures each, would respectively require
tables of 9, 16 or 25 sections.
90 THE COMBINATION TABLE.
THE COMBINATION TABLE.
99 x 99
TENS. UNITS.
81 | 72 | 63 | 54 | 45 | 36
72 | 64 | 56 | 48 | 40 | 32
63 | 56 | 49 | 42 | 35 | 28 21 | 28 | 35 | 42 | 49
54 | 48 | 42 | 36 | 30 | 24 | 18 | is | 24 | 30 | 36 | 42 | 48 | 54
45 | 40 | 35 | 30 | 25 | 20 | 45 | 15 | 20 | 25 | 30 | 35 | 40 | 45
36 | 32 | 28 | 24 | 20 | 16 12 | 16 | 20 | 24 | 28 | 32 | 36
a7 | 24 | 21 | 18 | 15 | 12 | 09 09 | 12 | 15 | 18 | 21 | 24 | 27
18 | 16 | 14 | 12 | 10 | 08 | 06 | | 06 | 08 | 10 | 12 | 14 | 16 | 18
09) 08) 0'7| 06) 05) 04 03 08) 04) 05) 06} 07; O8 O9
03 | 04 | 05 | 06 | 07 | 08 | 09
06 | 08 | 10 | 12 | 14 | 16 | 18
09 | 12 | 15 | 18 | 21 | 24 | 27
12 | 16 | 20 | 24 | 28 | 32 | 36
15 | 20 | 25 | 30 | 35 | 40 | 45
18 | 24 | 30 | 36 | 42 | 48 | 54
21 | 28 | 35 | 42 | 49 | 56 | 63
24 | 32 | 40 | 48 | 56 | 64 | 72
54 | 63 | 72 | 81
27 | 24 | 21 | 18 | 15.| 12 | 09
36 | 32 | 28 | 24 | 20 | 16 | 12
) 45 | 40 | 35 | 30 | 25 | 20
54 | 48 | 42 | 36 | 30 | 24 | 18
63 | 56 | 49 | 42 | 35 | 28 | 21
72 | 64 | 56 | 48 | 40 | 32 | 24
81 | 72 | 63 | 54 | 45 | 36 | 27
i
HUNDREDS. TENS,
The method of operation is as follows: the
factors being, for example, 74x63. Two squares
of card-board or other material are so adjusted as
to inclose on the factor lines.
THE COMBINATION TABLE. 91
07 tens and 04 units horizontally,
06 tens and 03 units perpendicularly,
as is here represented in diagram.
ae
21 03 12
“02
07 06 05 04 08 02 O1f 01 02 03 04
49 06 o4
The point a being uppermost, the par- 21
2
tial products may be read, one in each 4212
corner of the rectangle, and the product is 24
perceived to be 4662.
In a table of nine sections, the partial products
of two 3-digit factors may be indicated by stretching
threads squarely across the table so as to intersect
the tens figure of each factor, and then adjusting
the two squares to the factor figures expressing
units and hundreds. The nine partial products
then appear, one in each corner and one at each
92 THE COMBINATION TABLE.
intersection of the threads with each other, and
with the sides of the inclosure. In the accompany-
ing diagram are thus shown the nine partial pro-
ducts of 361 x 427, the crossed threads being repre-
sented by dotted lines.
THE COMBINATION TABLE. 93
The point a being uppermost, the par- 21
: 7 0642
Ep rerodacts appear as here arranged for 121907
addition. 9402
04
In this diagram may also be seen an orderly
arrangement of the partial products of many lesser
factors, formed by dropping one or more digits from
the original numbers.
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