Tutorial Arithmetic

Around 1900, Mr W. P. Workman, MA, B.Sc, the headmaster of Kingswood School in Bath, UK, assisted by Mr R. H. Chope, BA, wrote a book called "The Tutorial Arithmetic". It was published in 1902 by the University Tutorial Press Ltd, in London, England. Despite the name, this textbook was intended for high school students; and much of what it contains would be presented as Algebra today, or even as Number Theory. To quote Mr Workman:

Formal completeness is perhaps beyond the reach of an English author whose soul is vexed perforce with Weights and Measures and the barbarous mechanism by which they are manipulated, who is condemned also by the tyranny of an examination system to waste the space at his disposal in elaboration of the details of Complex Fractions, Abstract and Concrete, and other subjects of similar inelegance and uselessness. Moreover, further to militate against completeness, there exists an extraordinary prejudice against the use of literal symbols which are supposed to eb the inalienable property of Algebra. The consequence is that while nobody doubts that the work in Ex. 2, p.89, for instance, is Arithmetic, yet it becomes Algebra in the eyes of many if we replace "cost of clock" by c, i.e. use one letter instead of eleven to denote the qunatity in question. A really scientific treatise on Arithmetic is impossible so long as this divorce between Arithmetic and Algebra continues.

The book contains, of course, long sections on doing arithmetic with pounds, shillings, and pence; inches, feet, yards, furlongs, and miles; pints, quarts, gallons, pecks, and bushels; and so forth. But there is also a chapter on the metric system in which its simplicities are praised, while the fact that

...half a year of school life is entirely wasted for every English boy in learning the arithmetical devices necessary for managing the "weights and measures" previously explained.

is bemoaned.

To continue the subversion, a scheme is suggested (p.249 of the 1906 edition) whereby the English [sic] currency might be decimalized by calling in the threepenny bit, penny, halfpenny, and farthing, replacing them by a "cent" worth a hundredth of a pound, a "penny" of 1/2 cent, and "two-mil" and "one-mil" coins. The sovereign, half-sovereign, crown, double-florin, half-crown, florin, shilling, and sixpence would be left completely unchanged. It may be argued that this scheme had (in a day when the farthing, 1/960 of a pound, was a coin of non-negligible value) some advantages over the scheme eventually used! By 1971, the smaller coins would have been unnecessary; but introducing the name "cent" for the £0.01 coin might have avoided some confusion among those used to the "old penny".

There are also, typically for the period, long sections on financial mathematics, explaining the distinctions between debentures, stock, shares, bonds "the Funds", and "Consols"; how to draw up an invoice; and the theory of interest. However, there is also a distinct "New Math" flavor to parts of the book. Early in the book, there is a fairly advanced section on prime numbers (including Euclid's proof that there are infinitely many primes), and the largest prime number then known is given:


There are sections on "scales of notation" (that is, arithmetic in bases other than 10), and congruences (modular arithmetic, including Fermat's Little Theorem). Most interesting of all, though, is the section of "Harder Exercises" at the end of the book. These are mostly presented in the form of arithmetic problems; but, like some problems on mathematics competitions today, the arithmetic is intractable unless one moves from the concrete to the abstract and uses algebra and number theory. These are here reproduced in their entirety. They are long out of copyright: enjoy!

Page 480, Q1..Q4
Page 481, Q5..Q21
Page 482, Q22..Q34
Page 483, Q35..Q48
Page 484, Q49..Q64
Page 485, Q65..Q78
Page 486, Q79..Q96
Page 487, Q97..Q103
Page 488, Q104..Q114
Page 489, Q115..Q125
Page 490, Q126..Q135

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