PIKE'S ARITHMETIC

Assistant Professor of Mathematics

In the year of our Lord 1786, and of the Independence of these United States the eleventh, the Presidents of Harvard, Yale, and Dartmouth, and the Governor of Massachusetts found themselves in agreement. The three Presidents had read the manuscript of "A New and Complete System of Arithmetic composed for the Use of the Citizens of the United States" by Nicolas Pike, Esq., Schoolmaster at Newbury-port, and each of the three realized that a new prophet had arisen in Israel. Two years later Master Pike published his "New and Complete System" and to his very considerable financial betterment inserted the testimonials of these four scholars.

President Willard of Harvard showed the effect of a sound training in economic and political science by writing: "We are happy to see so useful an American production, which, if it should meet with encouragement it deserves, among the inhabitants of the United States, will save much money in the country, which would otherwise be sent to Europe, for publications of this kind. President Stiles of Yale wrote a neat little note containing the phrases: "production of genius," "suitable to be taught in schools," "we,ll adapted even for the university instruction." And from the North Country: "Dartmouth University, A.D, 1786. "At the request of Nicolas Pike, Esq, we have inspected his System of Arithmetic, which we cheerfully recommend to the public as easy, accurate and complete. And we apprehend there is no treatise of the kind extant, from which so great utility may arise to Schools.

B. Woodward, Math, and Phil. Prof. John Smith, Professor of the Learned Languages. "I do most heartily concur in the preceding recommendation. J. Wheelock, President of the University."

Whereupon the Governor of Massachusetts dipped his quill in the well and with a profusion of commas and a caution befitting the First Gentleman of so great a Commonwealth wrote: "From the known character of the Gentlemen, who have recommended Mr. Pike's System of Arithmetic, there can be no room to doubt that it is a valuable performance; and will be, if published, a very useful one. I therefore wish him success in its publication.

James Bowdoin."

Certainly the book was new, most certainly it was complete, but that it was arithmetic is not immediately obvious. Yet, if we strip away what purports to be algebra, geometry, physics, trigonometry, mechanics, , astronomy, and navigation, there is left a core of arithmetic. About half of this core consists of rules, the remainder of problems to be solved by these rules. And what rules! Here is a fair sample, the "Rule of Three."

"Let that number, which is of the same name or quality as the number sought, be the third term; then, consider whether the number sought should be more or less than the third; if more, let the greater of the other two terms be the middle term, and the less, the first; but if the fourth number ought to be less than the third, then give the less the second place, and the greater, the first. The question being thus stated, the proportion will be; As the first term is to the second, so is the third to the fourth, or number sought.

The first and second terms must always be brought into one name, and the third into the lowest mentioned, then proceed as in the common method, by multiplying the second and third terms together, and dividing the product by the first, and the quotient will be the answer, in the same name as the third term was reduced into."

Now this "Rule of Three" is nothing but a misguided method of finding out how much 12 bananas cost if 6 bananas cost 10 cents. Strip the legal verbiage from the rule and it tells you to pick out two of the three numbers, multiply them together, and divide by the third, and there is a gallant but wholly unsuccessful effort to tell. you which two to pick. So much for the rule, now how were problems actually done in the schools? The pupils who desired to know the price of 12 bananas when 6 bananas cost 10 cents, chose at random two of these numbers, say and 10, their product was 60, divided by 12, result 5 cents. Then he looked to see if that was the answer. It wasn't. He tried again. There are only three ways of doing this, two wrong, one right. Hence arose the maxim: "If at first you don't succeed, try, try again," This is a typical rule from a book which the Presidents of Harvard, Yale, and Dartmouth recommended as "best calculated to lead youth, by natural and easy gradations," "easy, accurate and complete!," "suitable to be taught in schools—of utility to the merchant, and well adapted even for the University instruction." Visiting Committees on "Exhibition Days" were wont to propound such a question as "If hens cost 40 cents each, how much will 5 eggs cost at 2 cents an egg?"

Young Archimedes: "40 times 5 are 200, divided by 2 are 100. Answer, 100 cents or one Federal Dollar."

Committee: "Wrong."

Archimedes: "40 times 2 are 80, divided by 5 are 16. Answer, 16 cents."

Committee: "Wrong."

Archimedes: "5 times 2 are 10, divided by 40 is 1/4 Answer, 1/4 of a cent."

Committee: "Wrong."

Very small boy speaks up: "5 times 2 are ten. Answer, 10 cents."

Committee: "Right."

Archimedes, much perplexed: "Oh, I thought 'twas to be done by The Rule of Three!"

And then we have those, twin brethren of depravity, "Alligation" and "False Position." Without much of any light as to what Alligation is we are informed it is of two kinds, Alligation Medial and Alligation Alternate. Then follow long and complicated rules full of "simples," "compounds," and "ingredients," and some illustrative examples with a curious interweaving of lines which connect various pairs of numbers. In appearance this is impressive, but the idea seems to have been merely this. Suppose you want to mix beans and corn to make succotash. Say beans are 10 cents a quart and corn 16 cents. If you mix 3 quarts of beans with 2 of corn, Alligation Medial tells you how much the succotash is worth; Alligation Alternate tells you what proportion of beans and corn to mix to get a 12 cent succotash. The problems here are all easy; but not if you try to do them by the rules.

"Position is a rule, which, by false or assumed numbers, taken at pleasure, discovers the true ones required." Like Alligation, Position is of two kinds, Single Position and Double Position. Both are, of course, False. Now that I have made perfectly clear just what Position is, let us "Resolve a Problem by Double False Position."

"A lady purchased a piece of silk for a gown at 80 cents per yard, and lining for it at 30 cents per yard; the gown and lining contained 15 yards, and the price of the whole was $7.00. How many yards of silk were there?" A word of caution. This was in the, year 1788, and we must not attempt to guess the answer in the light of the fashions of today.

Now this is how it is done. We make a guess that she bought 6 yards of silk, and this 6we label Guess No. 1. 6 yards of silk would cost $4.80, and then she must have bought 9 yards of lining costing $2.70; this totals $7.50, which is .50 too much. This .50 we label Error No. 1. We try again. Suppose Guess No. 2 is 3 yards of silk. That would cost $2.40, and 12 yards of lining would be $3.60, total $6.00, $1.00 too little. This $1.00 we label Error No. 2.

Now these errors are respectively too much and too little, so we proceed as follows. We multiply Guess No. Iby Error No. 2 which makes 6.00; then we multiply Guess No. 2 by Error No. 1 which makes 1.50. Add these two numbers, making 7.50. Add the errors, making 1.50. Divide 7.50 by 1.50 and we get 5. That's the answer, 5 yards of silk. The most annoying thing about this process is that it works.

Tonnage of Vessels is a topic included in all arithmetics of that day, and Pike gives rules for computing the tonnage of Merchant-ships and Men of War. Then branching out he proposes to find the tonnage of Noah's Ark. In the first edition he assumes 1 cubit equal 1 foot, and from the Pentateuchal dimensions finds its "Burthen" as a Merchant-ship to have been 4736 80-95 tons and as a Man of War 4500 tons, lit is disappointing to add that he ventures no opinion as to the proper classification. These figures evidently aroused some protest from the clergy for in the second edition he allows 22 inches to the cubit and revises his "Burthens" to 29,188.6 tons as a Merchant-ship and 27,720 tons as a Man of War.

Such an important problem as the tonnage of Noah's Ark deserves a fuller treatment than I can give here, but I feel that I ought to mention one other text, and point a moral. In 1835 Benjamin Greenleaf published his famous "National Arithmetic," a legitimate descendant of Pike. Greenleaf defines all vessels as single-decked or double-decked, gives rules for the government tonnage of each and then asks "What is the government tonnage of Noah's Ark, admitting its length to have been 479 feet, its breadth 80 feet, and its depth 48 feet? Ans. 17,421 9-19 tons." Our student now finds himself in a difficulty. Shall he try the single-decked rule or the doubledecked rule? If the latter he finds he cannot get the right answer, if the former, he can. Hence he has not only solved the problem, but has made the important discovery that the Ark was single-decked!

But in the later editions there is a curious and significant change. Some theologian, with mathematical tendencies, pondering this deeply, turned to Genesis VI: 16 and read . . . with lower, second, and third stories shalt thou make it." "Aha, lower, second, and third stories! That means double-decked; B. Greenleaf Esq. shall hear of this." And apparently he did, for in the later editions the answer given is that obtained by the double-decked rule. Thus at an early date in the history of our republic the fundamentalist exercised control over the labors of the scientist.

There are also problems in a lighter vein. It is pleasing to find in Pike an old friend: "What is the difference between six dozen dozen, and half a dozen dozen?" and it is a fact that this was ancient even in 1788. Our author is a confirmed "end-to-end statistician," and wants to know how many times a waggon wheel would rotate between Newbury-port and Boston, and how many barley-corns would reach around the earth.

He abounds in information which, at least in 1788, was useful. Consider the problem: ". . . .A seaman has a gallon of Brandy in a bottle, which weighs 4 1/2 lb. Troy, out of water, and, to conceal it, throws it Overboard into salt water; Pray, will it sink or swim?" Although this problem is now one of purely academic interest I will add that the answer is "sink."

Here is a problem in "Alligation Alternate" which is at least frank: "Suppose I have 4 sorts of currants at Bd., 12d., 18d., 22d. per lb.; the worst would not sell, and the best were too dear; I therefore conclude to mix 120 lb. and so much of each sort, as to sell them at 16d. per lb.: How much of each sort must I take?"

Finally a rule and a definition:— "To find the momentum of a fluid: Rule Divide the square of the velocity by 64 and multiply by 62.5 lb. Avoird. for clean river water, by 63 lb. for dirty water, and by 64 for sea water." "Definition: A spheroid is a Solid body like an egg, only both its ends are the same."

The evils that plagued our fathers are no more. Gone now are Barter and Fellowship and Practice, gone are Tare and Trett and Cloff, the Rule of Three, Alligation and False Position. Gone are the firkins of soap, the punches of prunes, and the fothers of lead. Gone apparently without hope of recall are the anchors of brandy, the runlets of spirits, the tierces of perry, the pipes, butts, and tuns of wine; gone are the firkins and kilderkins of ale and the puncheons of beer—even milk in those days was sold by the beer quart. Gone are the tods, pottles, strikes, cooms, weys, and lasts. Gone are the Spanish milled dollars, the pistareens, the pistoles, moidores, johannes, and half joes.

All that is left is a calf-bound, water- stained book, badly worm-eaten. There was something prophetic about the worms that invaded the copy I own. One of these survived the recommendation of the Governor of Massachusetts, but, penetrating to the Table of Contents, withdrew in great distress after a brief meal on Alligation and the Gregorian Epact. Another aspirant for knowledge, working his way from the rear, ate one-third of the transverse axis of an hyperbola, "expunged an adfected quadratic" with apparent relish, but died while gauging a wine-cask.

From the Portico of Webster Hall