Arithmetic as It Was Taught 175 Years Ago in Providence

July 6, 2006

Book Review
Philip J. Davis

Practical and Mental Arithmetic, on a New Plan, in Which Mental Arithmetic Is Combined with the Use of the Slate. Fifty First Edition, revised and enlarged. By Roswell C. Smith, Daniel Burgess & Co., Hartford, Connecticut, 1839, 284 pages.

I came across this book recently while clearing out some shelves. I had picked it up some years back for $10 during a vacation in the Castine, Maine, area. Its pages, though yellowed, are not crumbling, and the original binding is holding. My principal difficulty was the small type in a considerable fraction of the text---so small that I could read it only with a strong magnifying glass.

My first reaction on seeing the title page was, Aha! "A New Plan." What could the old plan have been? It's pretty clear to anyone who knows even a smattering of the history of mathematics that the way arithmetic has been done and taught has changed constantly ever since (and perhaps before) the Babylonians were recording the price of onions on cuneiform tablets. Educational schemes called the "New Math" or the "New New Math" always have been and always will be around.

It didn't take me long to discover from the introduction that the New Plan was inspired by the ideas of the famous Swiss educator Johann Heinrich Pestalozzi (17461827), which puts the author, Roswell C. Smith, at the cutting edge of 19th-century educational theory. The Pestalozzian theory has been epitomized in the slogan "Head, heart, and hands." I cannot go into the theory here, nor can I say how closely Smith hewed the Pestalozzian line. The designation "Fifty First Edition" reveals that Smith's book was highly successful and made bundles of money for its author.

The next thing I noticed were the slews of discarded, archaic words, phrases, ideas. Alligation (methods for dealing with mixture problems); duodecimals (not what you might think if you had a dose of the New Math of the '50s). Latitude 42 13' 32" (merely dealing with such numbers is a forgotten skill); bissextile (leap year); pistareen (a silver coin of Spanish origin, still in circulation); Flemish ell (about 27 inches). I leave my readers to discover the meaning of aliquot, pipes (as in pipes of wine), compound subtraction and division, fellowships, the rule of three.

What about the content of Smith's book? Numbers are divided into the simple and the compound, with the compound numbers comprising the common or vulgar fractions and the decimal fractions. The manipulation of these abstract creatures is described by many "rules" (i.e., algorithms), often given in conversational format. Here's the rule for simple addition:

Q. How do you write the numbers down?
A. Units under units, tens under tens, &c. with a line underneath.

Q. At which hand do you begin to add?
A. The right.

Q. If the amount of any column be 9 or less, how do you proceed?
A. Set it down.

Q. If it be more than 9, what do you do?
A. I set down the right hand figure, and carry the left hand figure or figures to the next column.

How I sometimes wish for a user's guide to walk me through a modern product with as much clarity.

Smith devoted seventeen pages to the arithmetic of "Federal Money." (Currencies of the various states were apparently still around in the 1830s.) Topics covered in the many sections on money include currency exchanges, discounts, interest, and insurance.

Impressive and eye-opening are the prices of things in the 1830s: beef, 8 a pound; wine, $1.15 a gallon; tallow, 11 a pound; whips, $1.31 each; snuff, 41 a pound; hymn-books, 44 each.

Now those were the days when the dollar had some clout, when the country that would need a good 5 cigar had one for 2. I mention these prices because the book's numerous drill problems---many "for the slate," often with answers attached---are accompanied by numerous problems in applied arithmetic, i.e., word problems. Here are a few:

"At nine pence an hour what will two years six months 3 weeks, 6 days, 12 hours labor come to in pence?"

"Reduce 160/171 of a quart to the fraction of a bushel."

"A merchant, having resided in Boston 6.2678 years, stated his age to be 72.625 years. How old was he when he emigrated to that place?" (I have never computed my age to four significant figures. Have you?)

"If one pair of shoes cost $2.25 . . ." (Leather shoes, hand made obviously, must have been expensive in 1839. I recall shoes at this price when I was a kid.)

"At $2.255 a gallon, how many gallons of rum may be bought for $28.1875?" (Hmm. Rum was cheaper than gasoline is now. But would it be politically correct today to mention alcohol in a book for school children?)

Smith seems to have taken pleasure in complicated numbers. Well, so do some number freaks today. He posed the following problem: "Suppose that one cent had been put at compound interest at the commencement of the Christian Era, what would it have amounted to at the end of the year 1827?" While neglecting to specify the interest rate, he gives the answer as an exact 46-digit number. Well, the nights were long in Providence in the days before TV. Much can be inferred about life in the USA in the early 1800s from a simple arithmetic book.

On the whole, the thrust of applied arithmetic la Smith is toward buying and selling, rather than toward construction or technology. (As President Calvin Coolidge would say in the 1920s, "The business of America is business.") But once in a while, the non-commercial world intrudes, as in:

"It is supposed that the wars of Bonaparte, in 20 years, caused the death of 2,000,000 persons. How many was this per hour, allowing the years to contain 365 days, six hours." (A question that, alas, is easily updated.)

Material presented in an appendix includes the extraction of square and cube roots, arithmetic and geometric progressions, annuities at compound interest. No algebraic formulas intrude: The reader is given only rules of procedure. There are only three pages on mensuration, but ten pages on a system of bookkeeping for farmers and mechanics. I don't think that many cube roots were extracted in the simple school rooms of the 19th century.

So who was Roswell Chamberlain Smith (17971875)? An easy session with a search engine yielded the answer. A country boy, Smith was born in Connecticut. His desire for more education than was then available in rural Connecticut led him to Phillips Andover Academy and then to Yale. He matriculated, but lack of funds prevented him from graduating. He became a teacher, first at the Norwich (Connecticut) Academy, and later in Providence, Rhode Island, where he ran a private school.

Smith wrote his first arithmetic book in Providence; it was a bang-up success. It's no wonder that my copy of Practical and Mental Arithmetic carries a plug from the Providence School Committee, chaired by the Reverend Francis Wayland, president of Brown University: The School Committee adopted the book. Smith followed Arithmetic not only with many editions and variations, but also with geography books and grammars, and had one success after another. The income from his writing enabled him to retire from teaching, and he moved to Hartford, where he devoted his life to producing texts.


Reading Smith's Arithmetic and, in close succession, checking out some groceries at my supermarket inevitably led me to the vexed question of what should be taught today as elementary arithmetic, and how. The checkers seem to know how to scan, and how to add dollars and cents from the till to make up the change. Can they also subtract? Of course, they have additional knowledge: how to deal with food stamps, coupons, credit cards, personal checks, and the 10% reduction on Tuesdays for Senior Citizens. I assume that the market has provided instruction. But they can't always tell the difference between a head of lettuce and a head of escarole, neither of which is zebra-striped.

Are the clerks taught the commutative law of addition and subtraction so that they can scan the cans and packages knowing that the order in which they do so makes no difference? Are these matters intuitive, or do I have it backward? Perhaps the commutative law arose from 19th-century formalizers doing some marketing. I do know that when the register crashes, I must rapidly take all my items to another line.

I have expressed my views of the current generation of questionable elementary math texts in "The Language Police Knock on Math's Door" (SIAM News, Volume
37, Number 6, July/August 2004; But all things pass, and these miserable tendencies and all current complaints will become irrelevant. A hundred seventy-five years from now, mathematicians will look at today's texts and have a good laugh. Will snuff still be around? Will people use chips implanted in their brains to calculate compound annuities mentally?

Smith's Arithmetic was not the first English-language arithmetic book published in America: William Bradford's Arithmetick made Easie came out in 1710. The rare book library at Brown has eleven copies of various editions of Smith, all in deep storage. For those of my readers who make a hobby of collecting old textbooks, I'm putting my copy up for sale. Make me an offer in Federal Money.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at

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