tiplied. The master explains the definition of multiplication, but the scholar is not satisfied, and the master says: "Master. If I multiply by more than one, the thing is increased; if I take it but once, it is not changed; and if I take it less than once, it cannot be as much as before. Then, seeing that a fraction is less than one, if I multiply by a fraction, it follows that I do take it less than once." "Pupil.-Sir, I do thank you much for this reason; and I trust that I do perceive the thing." The use of counters had not disappeared in England and Germany before the middle of the seventeenth century. Various methods of finger reckoning have been developed, and are commonly found in the older arithmetics. The accompanying cut is from Recorde's 'The Ground of Artes,' 1558, and gives a general idea of this practice. According to Pliny the image of Janus or the Sun was cast with the fingers so bent as to indicate 365 days. Some have thought that Proverbs iii, 16, "Length of days in her right hand," alludes to such a form of expressing numbers. An interesting illustration is given by Leslie: "The Chinese have contrived a very neat and simple kind of digital signs for denoting numbers, greatly superior to that of the Romans. Since each finger has three joints, let the thumbnail of the other hand touch these joints in succession, passing up one side of the finger, down the middle, and again up the other side, thus giving nine marks applicable to the decimal notation. On the little finger these signify units, on the next tens, on the next hundreds, etc. The merchants of China are accustomed, it is said, to conclude bargains with each other by help of these signs, and to conceal the pantomime from the knowledge of bystanders. The Korean schoolboy carries to school a bag of ing-bones, each about 5 inches long, and somew1 ner than the ordinary leadpencil. A box of square sticks, 4 inches in length and about 1⁄2 inch square, called sangi, is used in a very ingenious fashion by the Chinese for the solution of algebraic equations. The form of reckoning board adopted in the Middle Ages has left some words and customs. Fitz-Nigel, writing about the middle of the twelfth century, describes the board as a table about ten feet long and five feet wide, with a ledge or border, and was surrounded by a bench, or 'bank,' for the officers. From this 'bank' comes the modern word bank as a place of money changing. The table was cov ered after the term of Easter each year with a new black cloth divided by a set of white lines about a foot apart, and across these another set which divided the table into squares. This table was called "scaccarium," which formerly meant chessboard, from which is the term exchequer, the Court of Revenue. CHAPTER II CALCULATION UNDER the term 'logistic' the Greeks treated what is now ordinarily termed computation or calculation, the latter word coming from a Latin word meaning 'pebble,' inasmuch as the reckoning was done with counters or pebbles. Calculation is the process of subjecting numbers to certain operations now to be defined. There are six fundamental operations in arithmetic, all growing out of the first. Formerly these were differently classified, sometimes as high as nine being considered, the other three being special cases or complications of the fundamental six. These six operations are divided into two groups, the direct operations, of which there are three, and the inverses, each of which has the effect of undoing one of the former three. DIRECT I. Addition. 2. Multiplication. 3. Involution. INVERSE 4. Subtraction. 6. Evolution. When one object is put with a group of like objects, forming thus a new group having one more object than the original group, the process is said to be that of addition, and is indicated by +. (This sign appears in a work by Grammateus in 1514, and in 1517 in a book by Gillis vander Hoecke. Thus, I apple added to 2 apples gives 3 apples, or with abstract numbers, 2+1=3. The objects or numbers added are called 'addends' or 'summands,' and the resulting group or number is the 'sum.' The ending end or -and, so common in mathematical terminology, is Latin present passive participle; in this case Additio. 12 vourse 7 loed I Albie fein zu addiren die quantitet eines nas stenrotidí aftrechen 13: mens/als.mit I: primamit prima/secunda mut fecüda/tertia mit tertia 2c. Vnd mañ braubeert hepde de quadratati coĉti 144 cher solche zeichen als ist meh:/vnd/m:n pliceert dyesen quad:ace meteen anderen coemt 1 die der/in welcher fein zu mercken drei Regel. Die Erst Regel. treet van I barr na multf 144 144 nude Dacrafts Wannein quantitet hat an beyden oren + Cirrationale. eder-fofol mann folche quantitet addien hin CDietration2!efcar 1.166 — epsch vanden werfis / als treckt Be su gefagt das zeychen oder- Facit as pai.. 14 p.-14. Tie ander Regel. I oft met+naer den 3 vank; }; TJeem wildi aftrecken & van Be refill: 3 I in der Shern quantitet+vnd in der vn Citem&ildi aftrection 14 & wan (4) —; coffe dern-/vnd übertrifft —/ so sol die onder Ke quantitet von der dbern subtrah rt werden/vn su dem übrigen feg-Seaber die vnder qui tem van+oft —— ban —— fubtraheert ende + van- - oft van addeert wide fubtractie so ver titer ift groffer/fe fubtrahir die kleinern vöder reals & adberrhic fun oft fubtraheerlie. groffern/rn zu dem das do bleitendist/seac als opu.-+-6N: 4p".-+N. Multiplicatie niden K; van ratíontàïen. Wildi multipliceren inden K. forrece bat ghy indet ftellen alle de nommers van emidei notu Ire alo & te inultipliceren met simpelen neminerfce| moet gladen nominer multipliceren nar de qualiteye Eo in der obgefarten quantiter würt funds 6: Als wildi multipliceren: Romet 4 so set 4 in - und in der vndern+/vnd¬übertrifft+anutipliceert met 16 cofe 144 hier we freet Rroft fanen : multipliceert 4 in haer feluen coche 16 fo fubtraher eins von dem andern/vndzumu 12 foc rectis ghemultipliceert met 4/want R9 brigen fdbacib-3Resaber/das die ender que foz dit multipliceert met 4 coemt 12 als vozent. tutet übertrifft die obern/so ziche eins von Seni Voudummultipluarë Bince 5 so multipliceert andern/vnd zu dem ersten ferze -- alo Fig. 15-FIRST USE OF PLUS AND MINUS SIGNS. Left Column a Page from Grammateus; Right, from Gillis vander Hoecke. addends is to be translated literally the 'being added' numbers. Addition is, in its simplest form, the putting together or uniting of two numbers; and all additions of this nature may be broken up into a series of repetitions of the fundamental process of increasing a number by unity. Thus, if it be desired to add 3 apples to 5 apples, it may |