to sey, length{e}, brede, and thiknesse. so þ{a}t nombre hath{e} .3. nombres to be brought forth{e} in hym. But nombre may be had{e} twies in nombre, for other it is had{e} in hym-self{e}, oþ{er} in a-noþ{er}. If a nombre be had{e} twies in hym-self, oþ{er} ones in his quadrat, þ{a}t is the same, þ{a}t a cubik{e} [*Fol. 54.] is, And is the same that is solide. And yf a nombre twies be had{e} in a-noþ{er}, the nombre is cleped{e} solide and not cubik{e}, as twies .3. and þ{a}t .2. makith{e} .12. Wherfor it is opyn{e} that all{e} cubik{e} nombre is solid{e}, and not {con}u{er}tid{e}. Cubik{e} is þ{a}t nombre þat comyth{e} of ledyng{e} of hym-self{e} twyes, or ones in his quadrat. And here-by it is open that o nombre is the roote of a quadrat and of a cubik{e}. Natheles the same nombre is not q{ua}drat and cubik{e}. Opyn{e} it is also that all{e} nombres may be a rote to a q{ua}drat and cubik{e}, but not all{e} nombre quadrat or cubik{e}. Therfor sithen þe ledyng{e} of vnyte in hym-self ones or twies nought cometh{e} but vnytes, Seith{e} Boice in Arsemetrik{e}, that vnyte potencially is al nombre, and none in act. And vndirstond{e} wele also that betwix euery .2. quadrat{es} ther is a meene p{ro}porcionall{e}, That is opened{e} thus; lede the rote of o quadrat into the rote of the oþ{er} quadrat, and þan wolle þe meene shew. [Sidenote: Examples of square roots.] +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Residuu{m} | | |0| || | | |4|| | |0| | || | | 0 | | +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Quadrand{e} |4|3|5|6||3|0|2|9||1|7|4|2|4||1| 9 | 3 |6| +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Duplum |1|2| | ||1|0| | ||2| |6| | || |[8]|[{19}]| | +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Subduplu{m} | |6| |6|| |5| |5||1| |3| |2|| | 4 | |4| +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ [Sidenote: A note on mean proportionals.] Also betwix the next .2. cubikis, me may fynde a double meene, that is to sey a more meene and a lesse. The more meene thus, as to bryng{e} the rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat of the lesse. [Headnote: Chapter X. Extraction of Square Root.] [Sidenote: To find a square root. Begin with the last odd place. Find the nearest square root of that number, subtract, double it, and set the double one to the right. Find the second figure by division. Multiply the double by the second figure, and add after it the square of the second figure, and subtract.]