For the sake of further comparison, three illustrations from works in Mr. Plimpton's library, reproduced from the _Rara Arithmetica_, may be considered. The first is from a Latin manuscript on arithmetic,[584] of which the original was written at Paris in 1424 by Rollandus, a Portuguese physician, who prepared the work at the command of John of Lancaster, Duke of Bedford, at one time Protector of England and Regent of France, to whom the work is dedicated. The figures show the successive powers of 2. The second illustration is from Luca da Firenze's _Inprencipio darte dabacho_,[585] c. 1475, and the third is from an anonymous manuscript[586] of about 1500. [Illustration] As to the forms of the numerals, fashion played a leading part until printing was invented. This tended to fix these forms, although in writing there is still a great variation, as witness the French 5 and the German 7 and 9. Even in printing there is not complete uniformity, {145} and it is often difficult for a foreigner to distinguish between the 3 and 5 of the French types. [Illustration] As to the particular numerals, the following are some of the forms to be found in the later manuscripts and in the early printed books. 1. In the early printed books "one" was often i, perhaps to save types, just as some modern typewriters use the same character for l and 1.[587] In the manuscripts the "one" appears in such forms as[588] [Illustration] 2. "Two" often appears as z in the early printed books, 12 appearing as iz.[589] In the medieval manuscripts the following forms are common:[590] [Illustration] {146} It is evident, from the early traces, that it is merely a cursive form for the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions. 3. "Three" usually had a special type in the first printed books, although occasionally it appears as [Symbol].[591] In the medieval manuscripts it varied rather less than most of the others. The following are common forms:[592] [Illustration] 4. "Four" has changed greatly; and one of the first tests as to the age of a manuscript on arithmetic, and the place where it was written, is the examination of this numeral. Until the time of printing the most common form was [Symbol], although the Florentine manuscript of Leonard of Pisa's work has the form [Symbol];[593] but the manuscripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like [Symbol][594] and [Symbol][594] or [Symbol],[595] more closely resembling our own. The first printed books generally used our present form[596] with the closed top [Symbol], the open top used in writing ( [Symbol]) being {147} purely modern. The following are other forms of the four, from various manuscripts:[597] [Illustration] 5. "Five" also varied greatly before the time of printing. The following are some of the forms:[598] [Illustration]