16687 16432 Duodecimals into Decimals. Septenaries into Quaternaries. 1 × 12 + 6 = 18 1 × 7 + 6 = 31 × 12 + 6 × 7 + 4 --- ---- 222 1133 × 12 + 8 × 7 + 3 ---- ----- 2672 22130 × 12 + 7 × 7 + 2 ------ ------- _Answer_ 32071 1021012 Owing to our division of a foot into 12 equal parts, the duodecimal scale often becomes very convenient. Let the square foot be also divided into 12 parts, each part is 12 square inches, and the 12th of the 12th is one square inch. Suppose, now, that the two sides of an oblong piece of ground are 176 feet 9 inches 7-12ths of an inch, and 65 feet 11 inches 5-12ths of an inch. Using the duodecimal scale, and _duodecimal fractions_, these numbers are 128·97 and 55·_e_5. Their product, the number of square feet required, is thus found: 128·97 55·_e_5 --------- 617_ee_ 116095 617_ee_ 617_ee_ ------------ 68_e_8144_e_ _Answer_, 68_e_8·144_e_ (duod.) square feet, or 11660 square feet 16 square inches ⁴/₁₂ and ¹¹/₁₄₄ of a square inch. It would, however, be exact enough to allow 2-hundredths of a foot for every quarter of an inch, an additional hundredth for every 3 inches,[58] and 1-hundredth more if there be a 12th or 2-12ths above the quarter of an inch. Thus, 9⁷/₁₂ inches should be ·76 + ·03 + ·01, or ·80, and 11⁵/₁₂ would be ·95; and the preceding might then be found decimally as 176·8 × 65·95 as 11659·96 square feet, near enough for every practical purpose. [58] And at discretion one hundredth more for a large fraction of three inches. APPENDIX IV. ON THE DEFINITION OF FRACTIONS. The definition of a fraction given in the text shews that ⁷/₉, for instance, is the _ninth_ part of _seven_, which is shewn to be the same thing as _seven-ninths_ of a unit. But there are various modes of speech under which a fraction may be signified, all of which are more or less in use. 1. In ⁷/₉ we have the 9th part of 7. 2. 7-9ths of a unit. 3. The fraction which 7 is of 9. 4. The times and parts of a time (in this case part of a time only) which 7 contains 9. 5. The multiplier which turns _nines_ into _sevens_. 6. The _ratio_ of 7 to 9, or the _proportion_ of 7 to 9. 7. The multiplier which alters a number in the ratio of 9 to 7. 8. The 4th proportional to 9, 1, and 7.