Levi L. Conant's 1896 study opens with a striking claim: the Maori of New Zealand were once thought to count by 11s, with simple words for 121 and 1331. This error, Conant explains, arose from a counting habit where one object was set aside for each ten, not from an undecimal base. The correction exemplifies the book's method: close scrutiny of numeral systems to distinguish genuine patterns from misinterpretation. Conant, a mathematician at Worcester Polytechnic Institute, draws on original sources and recent authorities, compiling what he believes is the most extensive collection of binary, quinary, and other systems then existing. He omits ordinal numerals, focusing solely on cardinals, and preserves the orthography of source languages without diacritical marks.

Senary Curiosities and Their Limits

Conant devotes careful attention to purported senary (base-6) systems, finding that most examples are isolated compounds within larger decimal or vigesimal frameworks. The Mosquito tribe of Central America, for instance, uses a quinary-vigesimal scale but forms 7 as matlalkabe pura kumi (6+1), 8 as 6+2, and 9 as 6+3, before resuming ordinary counting at 10. Similarly, the Pawnee sequence shows 7 as petkoshekshabish (2-6, i.e., second 6) and 8 as touwetshabish (3-6, third 6), yet the overall system is decimal. Conant argues these are “accidental variations” rather than evidence of a senary base, comparing them to Wallachian deu-maw (2-9) for 18, which does not imply a nonary scale. The Uainuma term for 7 stretches to aira-ettagapi-hairiwigani-apecapecapsi, a compound whose meaning Conant admits is unknowable. Such examples, he concludes, are “of no importance whatever” to the structure of a system as a whole.

The Binary Trace in Compound Numerals

Throughout the excerpts, binary compounding appears as a recurring structural device. The Mosquito numeral for 4 is wal-wal (2-2), and the Pawnee 7 and 8 are built as multiples of 6 (2×6, 3×6). Conant notes that the Marshall Islands sequence includes thil thino for 6, meaning 3+3, and rua-li-dok for 8, meaning 10−2. These formations reveal a tendency to decompose numbers into smaller, familiar components, often using 2 or 3 as building blocks. The binary principle is most explicit in the Mosquito scale, where 2 is wal and 4 is its reduplication. Conant’s method is to identify such patterns without assuming they indicate a full binary system; instead, he treats them as local solutions to the challenge of naming quantities beyond the fingers of one hand.

Vigesimal and Quinary Overlap

The Mosquito system is described as “quinary-vigesimal,” combining a base-5 substructure with base-20 counting. The word for 5, mata-sip, literally means “fingers of one hand,” and 10 is mata-wal-sip (“fingers of the second hand”). This bodily origin is a common thread Conant traces across cultures. The vigesimal component appears in the overall scale, though the excerpts do not show the higher numerals. Conant’s preface notes that his collections of quinary and other systems are “the most extensive now existing in any language,” reflecting a comparative approach that prioritizes empirical data over theoretical speculation. He acknowledges assistance from anthropologists Horatio Hale and Frank Hamilton Cushing, grounding his work in contemporary ethnographic scholarship.

Conant’s study is best approached as a compendium of numeral systems rather than a narrative history. Readers will find detailed word lists and etymological notes that reward slow reading. The author’s caution—refusing to infer complete systems from isolated compounds—offers a model of scholarly restraint. Those interested in the cognitive foundations of mathematics will appreciate how Conant documents the inventive ways cultures have mapped numbers onto language, from finger-counting to the elusive senary curiosities that, upon inspection, dissolve into larger patterns.