S. I. Shubin.

Replacement for the Slide Rule. Type A.

An ordinary slide rule at its beginning makes it possible to read off with an accuracy of up to 2 units of the fourth digit, assuming that the eye can estimate 0.1 mm.; at the end, 2 units only of the third digit will already be distinguishable. The relative accuracy is thus \(1/500\). Hence the third digit begins to falter already from the middle of the rule. For example, in drawing up a monetary estimate this will already show in kopecks in sums not reaching 10 rubles. This circumstance considerably narrows the range of use of the slide rule.

On the other hand, its high price also hinders its wider dissemination, for example among technicians and students.

There are, it is true, Prelle calculating plates—one of them transparent. I once had such a pair in my hands—and the transparent one had already shrunk by several millimeters.

It is necessary to make both plates of the same material. There are also special calculating rollers, but these are already a whole machine. Accordingly, their price is also substantial. Engineer D. G. Ananov has invented a logarithmic plate. But, I think, calculation with it is rather difficult: with the aid of a rule with a runner one must take each number (for example, the multiplicand and the multiplier) and transfer it to a line whose number must be computed by a procedure similar to the computation of the characteristic, and moreover within the limits of 20.

In the proposed sheet, no computations have to be performed except the ordinary ones for characteristics (the number of digits before the decimal point), and even these are simplified.

In particular, our logarithmic sheet is convenient for powers, integral and fractional, and these powers are entering engineering more and more.

Description of logarithmic sheet A and reading.

The dimensions of the sheet have been chosen with the calculation that, despite all the crudeness of execution (simple printing), it would be possible even at the end of the sheet to have the fourth digit, obtaining it by visual division into 10 parts of 2 mm.

First, we see the primary scale—a series of divisions above the strokes (lines) with figures from 1.01 to 10.00. The distance between these lines is 1 cm. The end of the lower line corresponds to the beginning of the upper one. The scale is logarithmic. If it is printed on millimeter paper, then logarithms can be read off directly to the fourth digit: in each millimeter there are 2 units of the fourth digit. In each line there are 400 such units. If necessary, the calculator himself can make the ruling on the sheet.

With dividers in hand, using only this scale, one can already perform all computations doable on a slide rule, but one will have to count whole lines.

Under the primary scale, with small strokes downward, there is arranged a secondary, also logarithmic, scale, twice as small. Thus readings on the secondary scale are squares of the primary. And, starting from the secondary, on the primary we read off the square root.

But the secondary scale had to be repeated also in the intervals: the divisions intersect the horizontal strokes. Moving half a centimeter upward, we find the same reading on the same secondary scale to the left by half the length of a line; moving downward, one must step to the right. We shall call this latter scale the intermediate one.

For convenience in marking and reading, a decimal point has been introduced at the top (the decimal comma of decimal fractions). Thus the whole primary scale includes only the first decade, and each secondary one the first hundred.

At the side, in the margins, the numbers give the reading along the first and the last stroke with a digit on the scale, counting among them also those 9 little strokes for which only at the middle one stands the digit 5. At the very beginning of the sheet the numbers 1.01 refer to the second long stroke: the designation 1.00 would have said too little.

Another exception begins with the number 5.5 of the secondary scales up to the center of the sheet, and from 55 to the end: there the numbers in the margins indicate readings at the first and last strokes of the line. In addition to the size of the digits, the decimal point, which in the proper cases stands to the right or to the left of the digit, also helps one determine the place of the digit of the scale; if it is absent, then the digit of the scale denotes hundredths, and the further reading gives thousandths and ten-thousandths (by eye). In conclusion, a small remark: in reading, one or two zeros appear if the result has come out to the right immediately beyond a long stroke with a digit.

There are also enclosed 2 printed rulers with equal divisions and a zero in the middle, and another with a zero at one third of the length from the end; the digits on the other side of the zero are placed at double distances. These rulers should be carefully cut out with a knife. Gluing them onto anything is not recommended: the paper is sufficiently thick, and, on the other hand, on an uneven table it is better if the ruler follows the possible bends of the logarithmic sheet.

Multiplication and raising to a power.

1) \(a \cdot b\). We find the numbers \(a\) and \(b\) on the primary scale. It is useful, in all further operations as well, to stick pins into the places found: it is hard to think that, with a length, for example, of the primary scale of 5 meters, the logarithmic sheet will soon wear out.

With the aid of the enclosed ruler we divide the distance between the “points” \(a\) and \(b\) in half (so we shall call the places of the readings of the numbers \(a\) and \(b\)).

On the secondary scale—the lower or the intermediate one—we directly read off the product opposite the middle. When the distance between the lines is small and the ruler is almost vertical (from oneself), its intersection with the middle line gives, with great accuracy and clarity, the midpoint of the length \(ab\). When the distance between the points \(a\) and \(b\) is considerable, we see to it that the readings at the points \(a\) and \(b\) are identical—there is of course no need to memorize them for long. Then the product will be at the zero of the ruler (with equal divisions) on the secondary scale. Especially when the ruler is in positions close to horizontal (from left to right), an estimate by intersection would introduce greater errors*).

*) For more rapid division in half, one may recommend rubber bands (of various lengths) with equally spaced marks, or a set of equally spaced parallel lines on a transparent plate—changing the inclination. And on the enclosed rulers it is useful to color the corresponding sections with the same color.

Rationale. Since every procedure in work is more easily remembered and carried out if its idea is clear, I shall also give the rationale of the method. Take the numbers \(a\) and \(b\). To the midpoint of the distance on the primary scale between the points \(a\) and \(b\) from the beginning of the scale there corresponds the length \(\frac12(\lg a+\lg b)\) in the same units in which \(\lg a\) and \(\lg b\) were laid off. On the same primary scale we here read off \(\sqrt{ab}\).

It remains only, alongside the numbers already present on the scale, to place also their squares, which is what has been done on the secondary scale.

All is well if the midpoint of \(ab\) falls exactly on a line with the primary scale. But it may lie midway between the lines. For this case, in the interval between the lines the secondary scale is again taken, as if shifted to the left by half a line in comparison with the lower lines.

Position of the Decimal Point

a) It is easy to see that the secondary scale is repeated again from the middle of the sheet.

Since on the primary scale the graduation has been made in the first decade, it is natural in the second half of the secondary scale to carry the graduation up to one hundred, so as to read products (and squares) of numbers of the primary scale without thinking about the position of the decimal point.

If, however, numbers with a different number of digits before the point are multiplied, this must be taken into account in tens, hundreds, etc. For example, if the multiplicand and the multiplier are both 10 times larger (two-digit), then the product from the sheet must be increased 100 times.

b) For persons who have already worked with a slide rule: one must add the characteristics of the factors—and this is the characteristic of the result. If the product turns out to be in the upper half of the sheet, including here also the second half of the last of the lower-half intermediate lines, then one must add one more unit.

Or one must add the number of digits before the point (on the left)—and we obtain the number of digits of the result. If the result is in the lower half of the sheet, one should subtract one unit. The number of zeros after the point (on the right) we count as a negative number.

2) \(a^2\). Having found the number \(a\) on the primary scale, we read \(a^2\) below the line on the secondary scale. If in \(a\) the decimal point has shifted by several digits in comparison with the number on the sheet, then in the result it must be shifted in the same direction twice as far. Or see item 1) b).

3) \(a^3\) \((a^6)\). If the number \(a\) is only a little greater than one, then, placing on it the zero of the ruler with unequal divisions, we turn its short end until it coincides with the beginning of the scales (point A). Then the same reading along the ruler on the other side of zero will give \(a^3\) on the same primary scale: each coordinate and the whole logarithm have increased threefold.

But with a further increase of the number, its cube, with this procedure, will already go beyond the limits of the logarithmic sheet. In order for the time being to obtain only the digits of the cube (we shall clarify the position of the decimal point later), we proceed as follows: we divide, at least mentally, each side of the sheet into 3 equal parts. By means of such divisions we form, with vertical and horizontal lines, 9 equal rectangles. To find the cube, we apply the same method with the nearest point (instead of A) E, B, F, G, H, C, K, or D.

Thus in the central rectangle (say the fifth) we take the central point, while in the remaining ones, either at the side or nearԽ

angles. It is hardly necessary to outline these rectangles: an error is possible only in limiting cases, and even then it is immediately detected.

The reading on the secondary scale will give \(a^6\).

Justification. In view of the similar calculations, we shall carry it out only for the example at the point K.

Taking the reading on the primary scale at the point K, one must set \(10\).

The distance traveled along the scale from \(a\) to K is equal to \(\lg \frac{10}{a}\).

In accordance with the meaning of the method, we triple this distance and subtract it from \(\lg 10\):

\[ \lg 10 - 3 \lg \frac{10}{a} = \lg \frac{a^3}{100}. \]

Thus, on the scale we read not \(a^3\), but \(\frac{a^3}{100}\), which can also be used for determining the position of the decimal point.

Position of the decimal point. As a result, one must triple the number of digits before the decimal point or the number of zeros after the decimal point. If \(a\) (on the sheet) was less than \(2.1545\) \(\left(\sqrt[3]{10}\right)\), one must move the decimal point a further 2 places to the left (decrease by a factor of 100). If \(a\) was less than \(4.6416\) \(\left(\sqrt[3]{100}\right)\), one must move the decimal point a further 1 place to the left. In the first case the height of the point \(a\) is less than \(1/3\) of the entire height; in the second case the point \(a\) will lie in the middle band.

b) One can also indicate a somewhat more accurate method and, perhaps, a less complicated one: we align the number and the zero of the ruler with the unequal divisions and turn it up to the point A, with the end bearing the large divisions reversed. The same reading at the other end of the ruler coincides on the secondary scale with the reading \(a^3\). If this point threatens to go beyond the limits of the sheet (this is possible only in the upper or right third of the sheet and is easily detected), then instead of A one must take the points B, C, or D, of course the nearest one. And in the middle of the sheet, one number may serve for this purpose for 2, and even all 4, points A, B, C, D.

Justification. If the result were read on the primary scale, we would find \(a^{3/2}\), while the secondary scale gives squares, etc.

Position of the decimal point.

a) In finding the number on the sheet, we have shifted the point. Accordingly, in the result read on the sheet, one must shift the decimal point back three times as far. Only, if the points C or D were used, the result must additionally be increased by a factor of 10.

b) One must triple the characteristic. If the result falls in the upper half of the sheet, one must add 1 more. And, finally, if it was necessary to use the points C and D, then one should also add 1 to the characteristic of the result.

Or, one must triple the number of digits. If the result is in the lower half of the sheet, subtract 2; in the upper half, subtract only 1. If the points C and D were used, add an additional 1. The number of zeros after the decimal point is negative.

4) \(a^4\). The fourth power is even much simpler to read off. First, one may, of course, square twice according to item 2).

Secondly, with the aid of a ruler with equal divisions, double the distance from point A, B, C, or D—from the nearest one—which we easily find by dividing the whole sheet by eye into 4 equal rectangles. To double the distance, we place the zero at the point, and count off, for example, at A; with the same count from the other end of the ruler there will be the point \(a^4\) on the secondary scale: on the primary scale we would have obtained only \(a^2\).

Position of the decimal point. As a result, one must quadruple the number of digits before the point or the number of zeros after the point. If \(a\) (on the sheet) was less than \(1.7824\) \(\left(\sqrt[4]{10}\right)\), the point must be shifted another 3 places to the left.

For \(a\) less than \(3.1630\) \(\left(\sqrt[4]{100}\right)\), the point is additionally shifted 2 places to the left; and, finally, for \(a\) less than \(5.6235\) \(\left(\sqrt[4]{1000}\right)\), the point is shifted only 1 more place to the left.

5) \(abc\). The multiplication of 3 numbers can be carried out, first, successively. Only, having obtained \(ab\) on the secondary scale, it must again be transferred to the primary one. For searches from the lower half, one must double by eye the distance from A or B; for the upper half, one must take points C or D.

Secondly, for rapid and at least approximate orientation, we take the center of gravity of the triangle \(abc\), and cube the number at the center of gravity according to item 3) a) or b), without being disturbed by the fractional ordinate: the result must fall exactly on the primary scale, or on the secondary one in item 3 b).

This is the essence of the method. The details are as follows: according to item 1) we multiply \(a\) by \(b\). We obtain a definite point \(ab\). With the ruler with unequal divisions we connect it with point \(c\). We take one third of the distance from point \(ab\). This will be the center of gravity of triangle \(abc\). The order of operations is immaterial, and this can be used in the direction of simplification.

Justification. For example, the ordinate of the center of the triangle is expressed as:

\[ \frac{a_1+b_1+c_1}{3} \]

where \(a_1, b_1, c_1\) are the ordinates (row numbers) of the separate vertices. It remains to triple both coordinates.

Position of the decimal point. We add the number of digits before the point. The number of zeros after the point must be counted as negative.

Only if the center of gravity lies below \(1/3\) of the height of the whole sheet must the result be reduced 100 times. In the middle strip the result must also be reduced 10 times. In borderline cases one must see whether the first significant digit in the result has crossed over a unit, for example, in method 3 b).

6) \(abcd\). The multiplication of 4 numbers can, of course, be carried out successively, or, what will be somewhat faster, one may obtain \(ab\), then \(cd\), and so on.

But let us indicate a still faster way: we obtain \(ab\) and \(cd\), but do not read them off; instead, by the ordinary method we find the midpoint between the points \(ab\) and \(cd\) (on the secondary scales), and, finally, use the second method from item 4), without being disturbed by the fractional ordinate: all the same, the result will fall on the secondary, even if intermediate, scale, from which the result must be read.

Position of the decimal point. We add the number of digits before the point. The number of zeros after the point is counted as negative.

Only, if the point \(\sqrt[4]{\mathrm{abcd}}\) lies below \(1/4\) of the height of the whole sheet, the result must be reduced 1000-fold. If the point falls in the second quarter of the height (the second band), the result must be reduced 100-fold. And, finally, by 10-fold if the point \(\sqrt[4]{\mathrm{abcd}}\) falls in the third band. In doubtful cases attention must be paid to the first digit: if the number obtained begins with 1, the reference band should be assigned to the older of those assumed; but if it begins with 8 or 9, then to the younger of those assumed.

Division.

7) \(\dfrac{10}{a};\ \dfrac{1}{a}\). For division of 10 by \(a\), we place the zero of the ruler with equal divisions at the central point and by rotation bring it to the point \(a\) on the primary scale. On the other side of the zero, at the same distance, on the primary scale itself, we read off the required value (on the secondary scale, \(\dfrac{100}{a^2}\)); if one proceeds from the secondary scale to the primary, then by the indicated method we find

\[ \frac{10}{\sqrt{a}}. \]

What has been said applies fully to the number \(a\) read on the sheet. We shall find the place of the decimal point below.

It is clear that this same method gives the digits of the reciprocal number \(\left(\dfrac{1}{a}\right)\). For the place of the decimal point we propose the following rule: the number of digits before the point is equal to the total number of zeros (counting also the zero on the left) before the first significant digit in the reciprocal number; a number with one digit before the point gives a reciprocal with only one zero (to the left of the point). Increasing the number by 10, 100, . . . times, conversely, decreases the reciprocal by 10, 100 . . . times. This rule may be useful in division.

8) \(\dfrac{a}{b}\). First of all, the following method suggests itself: first find the reciprocal of the divisor, and then multiply. But this requires two operations with the ruler.

For division in a single operation, however, one must proceed in the reverse way as compared with multiplication. Namely, we first find the dividend on the secondary scale.

At this point we place the zero of the ruler with equal divisions. We look to see with which division of the ruler the divisor coincides. Finally we find the quotient on the primary scale opposite the same division of the ruler on the other side of the zero. When the ruler is in a position close to vertical, the accuracy of the readings plays little role: what is important is the point of intersection with the required line. All this has been explained in more detail in the chapter on multiplication.

Only the other end of the ruler may turn out to be outside the sheet. To avoid this, we proceed as follows: in the dividend and in the divisor we separate off only one digit on the left—if the dividend turns out to be greater than the divisor, then the dividend must be sought on the secondary scale in the lower half of the sheet. In the opposite case, we increase the dividend 10-fold and find it in the upper half of the sheet. As necessary, of course, one may at times have to take the number also on the intermediate secondary scale (half a line to the right or to the left of the double scale).

Place of the decimal point. a) It is necessary to count how many times we have increased or decreased (by 10, 100, . . .) our numbers when we were finding them on the sheet, etc.

b) From the number of digits before the decimal point in the dividend one must subtract the number of digits of the divisor. If the dividend was taken on the lower half of the sheet, one must further add one. The number of zeros after the decimal point plays the role of a negative number.

When subtracting characteristics, we subtract another 1 if the dividend is taken on the upper half of the sheet.

Combination of Multiplication and Division

8) $\dfrac{ab}{c}$. In such a formula it is simpler to begin with the division: the intermediate result will fall on the primary scale, and it need not be read off, but only marked.

9) $\dfrac{abc}{de}$ and $\dfrac{ab}{cd}$. In these complex cases also it is simpler to begin with division and alternate with multiplication. Sometimes, of course, the second result will have to be shifted by half a sheet sideways or upward (downward).

10) $\dfrac{a^2}{b}$. We find $a$ on the primary scale. On the secondary scale, as chance may have it, there stands $a^2$, i.e. the dividend for the division may turn out to be just this point. But it is now also easy to seek the suitable point.

If the points $a$ and $b$ are close to one another, the following procedure may also be tried: place the zero of the ruler with equal divisions at the point $a$; in the direction opposite from $b$, move along the ruler an equal distance. Carry everything out on the primary scale.

Justification. The distance along the scale between $a$ and $b$ is $\lg \dfrac{a}{b}$.

11) $\dfrac{a^3}{b}$. If we begin according to item 10), the point $a$ will be marked for us.

It remains only to return to it. Or see item 3 b).

Extraction of Roots

12) $\sqrt a$. Our sheet also constitutes a table of square roots (and squares).

Let us recall the usual division into periods of 2 digits when extracting the square root. If in the first (left) period there are 2 digits, then we look for the number on the secondary scale in the upper half of the sheet, and above it, on the primary scale, read the root.

With the decimal point we separate off as many digits on the left as there were periods before the point. In the case of a proper decimal fraction, after the point we put as many zeros as there were complete periods with zeros (pairs) after the point (before the periods with significant digits). One may also apply item 13) (the following).

13) $\sqrt[4]{a}$. Here likewise we divide the number into periods of 4 digits. If in the first period there are 1 or 3 digits, we take the number on the lower half of the sheet on the secondary scale. In the case of one digit in the 1st period, we halve the distance between our point and A or B, so that the result falls on the primary scale.

In the case of 3 digits in the first period, we find the midpoint of the segment joining our point with C or D. We find the result on the primary scale.

If there are 2 or 4 digits in the first face, we take the number on the upper half of the sheet and then proceed as before, in the same order.

With the point we separate off as many places (zeros) as there are faces before the point (faces of 4 zeros each in the case of a proper decimal fraction).

If, however, we take the number \(a\) on the primary scale, then we extract only the square root. We use the points C or D when there are 2 significant digits in the first face.

14)

\[ \sqrt[3]{a}\cdot\left(\sqrt[6]{a}\right). \]

We divide the number into faces of 3 digits each. The decimal point, as in the preceding cases, must lie between faces.

a) If the first face has one significant digit, we connect our number on the primary scale with the points A, E, or B and take the first third of the whole distance from one of the latter points (with the aid of a ruler with unequal divisions), and we also read off on the primary scale. To make it easier to find one’s way in choosing the points A, E, or B, the lines of the primary scale have also been renumbered.

In the case of 2 digits in the first face we use the points F, G, or H. And finally, for a full first face one must take the point C, K, or D.

With the decimal point we separate off on the left as many places (zeros) as there were faces on the left (faces with zeros on the right).

b) One may also use the method of item 3 b). Namely, if there are 1 or 2 digits in the first face, we find the number on the corresponding secondary scale; we connect it with the point A or B and take the point at one third of the distance from the number (at \(2/3\) from A or B). We read off on the primary scale. In order that the third should fall precisely on the primary scale, it is necessary to choose A or B, while the secondary scale may, perhaps, be taken in the interval. To make orientation easier, the rows of the primary scale have also been numbered at the side: the number of centimeters or halves vertically (from A or B) must be divisible by 3. If there are 3 digits in the first face, we take the number on the upper secondary scale and use the point C or D. In this case one may remember that before extracting the root we also multiply by 10.

To S. I. Shubin’s article: Replacement for the Slide Rule. Type “A”.

LOGARITHMIC SHEET

(Large logarithmic sheet diagram with scales labeled \(C\), \(K\), \(D\), \(A\), \(E\), and \(B\).)

(At right: a separate vertical rule labeled “Rule for the logarithmic sheet.”)