Augustus De Morgan opens his Elements of Arithmetic with a frank declaration: this is not a book for those who want mere routine. The preface, dated 1846, positions the work within a pedagogical shift toward 'rational arithmetic'—teaching the reasons behind rules, not just the rules themselves. De Morgan's authorial voice is precise and occasionally wry, as when he notes that the measures used in England are not those a scientifically informed people would have chosen. Throughout, he insists that understanding, not speed, is the goal.

A Preface That Sets the Tone

The preface is unusually direct for a textbook. De Morgan warns that 'those who want nothing but routine' should look elsewhere. He traces the book's evolution from 1830 to the fifth edition of 1846, noting that the body of the work remains largely unchanged but that eleven new appendixes have been added. The first appendix, on computation, and the sixth, on decimal money, he urges every student to 'read and practise … with as much attention as any part of the work.' This is not a boast about completeness but a practical directive. De Morgan's tone is that of a teacher who expects effort and rewards it with clarity.

Reasoning With Counters and Combinations

De Morgan frequently uses concrete objects—counters, yards, feet—to ground abstract operations. In the section on permutations and combinations, he explains that selecting seven counters out of ten leaves a combination of three, so the number of combinations of seven equals that of three. He then shows the algebraic cancellation: the formula for seven out of ten reduces to the formula for three out of ten. The exercises that follow are stark: 'How many combinations of four can be made out of twelve things? Answer, 495.' The final example—the number of possible whist hands from a 52-card deck—yields 635,013,559,600, a figure that astonishes without commentary. De Morgan lets the numbers speak.

The Practical Problem of Measures

Book II opens with a meditation on units. De Morgan imagines comparing two room lengths expressed as fractions of a mile: 1/180 and 1/174. He notes that such fractions give 'no idea' of the difference. Converting to yards yields 9 7/9 and 10 10/87—still imperfect. Subdividing the yard into feet produces 9 yards, 2 feet, 1/3 foot versus 10 yards and a bit over 1/3 foot. The point is not the specific numbers but the principle: multiple measures exist for convenience, not necessity. De Morgan then remarks that England's system is not the one a scientifically informed people would have devised—a quiet critique embedded in a textbook.

The Epigrams That Frame the Work

Two epigrams appear on the title page. The first, from Melanchthon, asserts that 'the beginning is half of the whole'—especially true in philosophy. The second, from Condillac, argues that one learns not by routine but by reflection, and that the habit of reasoning about what one does is 'acquired more easily than one thinks' and once acquired, never lost. These quotations are not decorative; they encapsulate De Morgan's entire pedagogical stance. The book is designed to instill that habit. The epigrams also reveal De Morgan's intellectual lineage: he aligns himself with humanist and Enlightenment thinkers who valued method over memorization.

De Morgan's Elements of Arithmetic rewards the reader who works through its examples and reflects on its reasoning. The appendices—particularly those on computation and decimal money—offer practical skills that were forward-looking in 1846. A reader coming to this text should expect to think, not just to calculate. The book is best approached as a dialogue with a teacher who values precision and understanding above all.