Set 8·6 to index and pointer to 3·5. Bring 0·73 to pointer (noting
that 1 on the scale passes the index) and set pointer to 0·23. Set
1·02 to pointer (noting that 1 on the scale passes the pointer) and
read under index 7·95. There are 1 + 0 + 1 = 2 numerator digits and 1
+ 0 = 1 denominator digit; while 1 is to be added and 1 deducted as
per rule. But as the latter cancel, the digits in the result will be 2
− 1 = 1.
When moving the dial to the left will cause 1 on the dial to pass _both_
index and pointer (thus cancelling), the dial may be turned back to make
the setting.
It will be understood that when 1 is the _first_ numerator, and 1 on the
dial is therefore set to the index, no digit addition will be made for
this, as the actual operation of calculating has not been commenced.
In the Stanley-Boucher calculator (Fig. 23) a small centre scale is
added, on which a finger indicates automatically the number of digits to
be added or deducted; the method of calculating, however, differs from
the foregoing. To avoid turning back to 0 at the commencement of each
calculation, a circle is ground on the glass face, so that a pencil mark
can be made thereon to show the position of the finger when commencing a
calculation.
_To Find the Square of a Number._— Set the number, on one or other of
the square root scales, to the index, and read the required square on
the ordinary scale.
_To Find the Square Root of a Number._—Set the number to the index, and
if there is an _odd_ number of digits in the number, read the root on
the inner circle; if an even number, on the second circle.
_To Find the Cube of a Number._—Set 1 on the ordinary scale to the
index, and the pointer (on the back dial) to the number on one of the
three cube-root scales. Then under the pointer read the cube on the
ordinary scale.
_To Find the Cube Root of a Number._—Set 1 to index, and pointer to
number. Then read the cube root under the pointer on one of the three
inner circles on the back dial. If the number has
1, 4, 7, 10 or −2, −5, etc., digits, use the inner circle.
2, 5, 8, 11 or −1, −4, etc., „ „ second circle.
3, 6, 9, 12 or −0, −3, etc., „ „ third circle.
_For Powers or Roots of Higher Denomination._—Set 1 to index, the
pointer to the number on the ordinary scale, and read on the outer
circle on the back dial the mantissa of the logarithm. Add the
characteristic (see p. 46), multiply by the power or divide by the root,
and set the pointer to the mantissa of the result on this outer circle.
Under the pointer on the ordinary scale read the number, obtaining the
number of figures from the characteristic.
_To Find the Sines of Angles._—Set 1 to index, pointer to the angle on
the outer circle, and read under the pointer the _natural sine_ on the
ordinary scale; also under the pointer on the outer circle of the back
dial read the _logarithmic sine_.
THE HALDEN CALCULEX.—After the introduction of the Boucher calculator in
1876, circular instruments, such as the Charpentier calculator, were
introduced, in which a disc turned within a fixed ring, so that scales
on the faces of both could be set together and ratios established as on
the slide rule. Cultriss’s Calculating Disc is another instrument on the
same principle. The Halden Calculex, of which half-size illustrations
are given in Figs. 24 and 25, represents a considerable improvement upon
these early instruments. It consists of an outer metal ring carrying a
fixed-scale ring, within which is a dial. On each side of this dial are
flat milled heads, so that by holding these between the thumb and
forefinger the dial can be set quickly and conveniently. The protecting
glass discs, which are not fixed in the metal ring but are arranged to
turn therein, carry fine cursor lines, and as these are on the side next
to the scales a very close setting can be made quite free from the
effects of parallax. This construction not only avoids the use of
mechanism, with its risk of derangement, but reduces the bulk of the
instrument very considerably, the thickness being about ¼in.
On the front face, Fig. 24, the fixed ring carries an outer
evenly-divided scale, giving logarithms, and an ordinary scale, 1–10,
which works in conjunction with a similar scale on the edge of the dial.
The two inner circles give the square roots of values on the main scales
as in the Boucher calculator. On the back face, Fig. 25, the ring bears
an outer scale, giving sines of angles from 6° to 90° and an ordinary
scale, 1–10, as on the front face. The scales on the dial are all
reversed in direction (running from right to left), the outer one
consisting of an ordinary (but inverse) scale, 1–10, while the three
inner circles give the cube roots of values on this inverse scale. As
the fine cursor lines extend over all the scales, a variety of
calculations can be effected very readily and accurately.
[Illustration: FIG. 24.]
[Illustration: FIG. 25.]
SPERRY’S POCKET CALCULATOR, made by the Keuffel and Esser Company, New
York (Fig. 26), has two rotating dials, each with its own pointer and
fixed index. The S dial has an outer scale of equal parts, an ordinary
logarithmic scale, and a square-root scale. The L dial has a single
logarithmic scale arranged spirally, in three sections, giving a scale
length of 12½in. The pointers are turned by the small milled head, which
is concentric with the milled thumb-nut by which the two dials are
rotated. The gearing is such that both the L dial and its pointer rotate
three times as fast as the S dial and pointer. All the usual
calculations can be made with the spiral scale, as with the Boucher
calculator, and the result read off on one or other of the three
scale-sections. Frequently the point at which to read the result is
obvious, but otherwise a reference to the single scale on the S dial
will show on which of the three spirals the result is to be found.
[Illustration: FIG. 26.]
_The K and E Calculator_, also made by the Keuffel and Esser Company, is
shown in Figs. 27 and 28. It has two dials, of which only one revolves.
This, as shown in Fig. 27, has an ordinary logarithmic scale and a scale
of squares. There is an index line engraved on the glass of the
instrument. The fixed dial has a scale of tangents, a scale of equal
parts and a scale of sines, the latter being on a two-turn spiral. The
pointers, which move together, are turned by a milled nut and the
movable dial by a thumb-nut, as in Sperry’s Calculator, Fig. 26.
[Illustration: FIG. 27.]
[Illustration: FIG. 28.]