# Were tables of square roots ever in use?

Before the advent of calculators they had useful ready made tables for the main functions:sines,cosines logs etc..., do you know if tables of square roots were ever produced or in use?

I never heard of it, yet, it would be very easy to produce, since it is enough to find the roots of numbers from 1 to 100.

• Tables of base-ten logarithms were used to faciliated calculations. The reason for using $10$ as the base is that if one wants, for example, the logarithm of $7314,$ one finds $7.314$ in the table, which goes only from $1$ to $10,$ and then one adds $3$ to the logarithm, corresponding to moving the decimal point three places to the right. $\qquad$ – Michael Hardy Jun 9 at 4:27
• One should add that there's an easy algorithm for square roots. So in a sense you did not need a square root table as much as you needed one for logs and trigs. Moreover, you can find roots with logs. – Chrystomath Jun 9 at 16:05
• Oh you sweet summer children.... (I continue to be amazed at what youngsters are not taught or exposed to) – Carl Witthoft Jun 10 at 12:28

Pretty much every mathematics textbook (school or college) before the early 1980s (and many even up to the late 1980s), at the algebra level or above, as well as many (most?) chemistry and physics and engineering textbooks, had such a table at the back of the book (as an appendix or something, where "selected answers" and "index" and "glossary" would appear). Often the entries would be for both $$\sqrt{n}$$ and $$\sqrt{10n},$$ which was enough to allow you to easily get approximations for any magnitude-size numbers. For example, to find an approximation for $$\sqrt{3880},$$ use $$n=3.88$$ and look in the $$\sqrt{10n}$$ entries, since

$$\sqrt{3880} \; =\; \sqrt{1000\times 3.88} \; = \; 10\sqrt{10 \times 3.88}.$$

There were stand-alone (i.e. as separate books) tables also, such as the following, where square roots from $$1.00$$ to $$9.99$$ by increments of $$0.01$$ are on pp. 16-17 and square roots from $$10.0$$ to $$99.9$$ by increments of $$0.1$$ are on pp. 18-19:

Mathematical Tables for Class-Room Use by Mansfield Merriman (1915) https://archive.org/details/mathematicaltabl00merrrich

When I was in high school I owned (purchased in 1974) and used the 20th edition (1973) of the CRC Standard Mathematical Tables. On pp. 71-90 you'll find a table having column entries for $$n^2$$ and $$\sqrt{n}$$ and $$\sqrt{10n}$$ and $$n^{3}$$ and $$\sqrt[3]{n}$$ and $$\sqrt[3]{10n}$$ and $$\sqrt[3]{100n}$$ from $$n=1$$ to $$n=1000$$ by increments of $$1.$$ In high school I also owned (I no longer seem to have it, however) Logarithmic and Trigonometric Tables to Five Places by Kaj L. Nelson (in the well known Barnes and Noble College Outline Series of books), and other people I knew (in college) had the Schaum's Outline Series of Mathematical Handbook of Formulas and Tables by Murray R. Spiegel (1968), which I didn't have a copy of back then but a few years ago I saw and purchased a copy of a later printing (the 1990 printing) at a local used bookstore (square roots are on pp. 238-239). However, in looking at the Schaum's book now, it's more of a handbook of formulas (algebraic, trigonometric, calculus, series, special functions, etc.) than a table of numerical values for computational use.

For other such books, try this search and similar searches. For tables at the back of textbooks, simple archive.org and google-books searches will give you hundreds (if not thousands) of examples where you'll find square root tables at the back of the book.

• When I was in high school (the early 70's), the CRC book was too expensive for me. I used (and still have) the Schaum Mathematical Handbook, but I mostly used a smaller book that was just tables. It was similar to your Logarithmic and Trigonometric Tables book but the cover was different. That was the book I used the most--so much that it fell apart on me a couple of years ago so I had to throw it away. And yes, I did use the table of square roots. +1 from me--thanks for the memories. – Rory Daulton Jun 8 at 9:39
• +1 ... I, too, have the CRC book (a prize for a mathematical contest). But now I am considered to be "history" I guess. – Gerald Edgar Jun 8 at 11:33
• @Roy Daulton: I was taking our 4th year math course (essentially trigonometry in the fall, and precalculus math including conics and logarithms and matrices and probability and math induction in the spring) during 1974-1975 (my sophomore HS year), and our teacher was able, I think, to get a special discount for us, so maybe 6 to 8 of us (out of around 30 total in the two 4th year classes) wound up getting a copy in fall 1974. For what it's worth, about 2 years ago, for prosperity purposes, I made a .pdf scan copy of my notes for that class (213 pages). – Dave L Renfro Jun 8 at 11:49
• @Roy Daulton (and Edgar): For more memories, see my answer to Using log table to solve a division problem. – Dave L Renfro Jun 8 at 12:15
• @user157860: The beginning of various editions of Charles Hutton's Mathematical Tables (1785 edition, other editions) have a lot of historical information. More complete historical information is known today, but it's interesting to read accounts written back then, which don't have present-day written myopic treatments in which everything is viewed through the lens of (electronic) computers. – Dave L Renfro Jun 8 at 17:49

The Babylonian clay tablet from around 1700 BC known as YBC7289, since it's one of many in the Yale Babylonian Collection has a diagram of a square with one side marked as having length 1/2. They took this length, multiplied it by the square root of 2, and got the length of the diagonal.

Given that square roots were in use then, it would have made sense to create a table of such, rather than repeating the labour in calculating the result.

• After seeing all these antiquities it seems that humans were quite intelligent thousands of years ago. From Mesopotamia to the building of huge pyramids---mysteries are spread everywhere. – M. Farooq Jun 8 at 16:00

In addition to the existence of such tables, (in the later 1960's, for example) high school math classes included some lessons on (linear) interpolation from the values given in tables.

And some of us wondered about higher-order interpolation, and so on, which does nicely lead to many topics in calculus ... and splines and other stuff.

Slide rules were a very good competitor to "tables", in many situations, since they could give quicker answers (tho' somewhat lower precision) and were more portable.