|Mini-Series On Classical Mathematics:|
As History of Mathematics ... Sigh ...
Absolute and Relative Roman Numerals
Two is Still Not a Number With any Square Root
|Now, for the essay:|
The first of such proofs in European mathematics appears in a treatise of 1344 by Maestro Dardi. It explains why a negative multiplied by a negative makes a positive. It is repeated in various other manuscripts dealing with algebra during the fifteenth century. The proof is derived from the well-know operations on binomials which often appear in the introductions of such manuscripts. The reasoning goes as follows: we know that 8 times 8 makes 64. Therefore (10 – 2) times (10 – 2) should also result in 64. One well known multiplication procedure is called per casella, (literally by the pigeonhole) meaning cross-wise multiplication in which you add all the sub-products (see Swetz 1987, 201-205). You multiply 10 by 10, this makes 100, then 10 times – 2 which is – 20 and again 10 times – 2 or – 20 leaves us with 60. The last product is – 2 times – 2 but as we have to arrive at 64, this must necessarily be + 4. Therefore a negative multiplied by a negative always makes a positive. In modern terminology we would say that the proof is based on distributive law of arithmetic.
Historical objections against the number line
by NN (a Belgian, I think)
I totally concede that (10-2) times (10-2) make 64. I also concede that the multiplication per casella makes room for partial results 100, - 20, - 20 and + 4. But I do not concede that multiplication by casella is essentially reflecting the mathematic reality. It is a device for ease of operations.
Cardano did not accept + 4 as a result of -2*-2, but as a construct, exactly the explanation I heard of my own mathematics teacher: draw a line a-(8)-b-(2)-c, draw from it downwards lines a-(8)-d-(2)-e, b-(8)-f-(2)-g and c-(8)-h-(2)-i. Join d-(8)-f-(2)-h in a line and e-(8)-g-(2)-i in a line. Cardano argues that the square f-h-i-g has to be added so it can be subtracted twice, once when you subtract oblong d-h-i-e, and once when you subtract b-c-i-g. However, nobody would physically do that. One would either start subtracting f-h-i-g (-4) and then subtract b-c-h-f (-16) and d-f-g-e (-16), or one would subtract first either b-c-i-g (-20) and then d-f-g-e (-16) or one would subtract d-h-i-e (-20) and then b-c-h-f (-16).
Therefore it cannot be a basic rule of arithmetic reality that -2*-2 = +4, but it is a construct either pertaining to the algorithm of multiplication per casella or to a very screwed explanation of the geometric equivalent, one only chosen - as is the multiplication per casella - for ease of calculation.
It remains true that a subtraction from what is to be subtracted, like -16 = -(20-4), though not per se a positive, is equivalent to a positive added in order to compensate for too much being subtracted. But this does not mean that true negative numbers (there is no such thing) yield if multiplied together truly positive products.
Some early examples, like Wallis, come from treatises of algebra. That is as it should, since algebra is the science fiction and fantasy section of mathematics. It is a shorthand for the real operations, hence it is a form of makebelieve.
Georges Pompidou BiP Library
St James' Day
PS: I was just writing about multiplication "per casella" above. Now, I was reading an assertion that it was virtually impossible to multiply CXV by VI. Of course not, one used equally the multiplication per casella. I will give an example of what this means in both cases. Write one factor detail by detail above and another similarily at the right of a grid. Then add up whatever is inside the grid. I will give the factors in bold, since I have little possibility to show the border of the grid:
|100 -20 -20 +4=64||DCLXXXVV=DCXXXX|
In the same book where I find this groundless assertion - typically enough one of the histories of learning and technology written by modern day to day users of "exact sciences" - I found also this about Pope Sylvester II:
Pope Sylvester II, who died in 1003, introduced the decimal system as we know it today. The ease with which calculations were possible all of a sudden could only be interpreted by medieval mind as work of the devil. Accordingly, the mortal remains of the pope were exhumed and his corpse exorcised.
P. 11 English column of Calculus, Les machines du calcul non électriques / Nietelektrische Rekenmaschines / Non-electric calculating machines by Luc de Brabandere (Univ. de Louvain) in editions Mardaga, Liège.
If the poor Pope Sylvester II was exhumed and exorcised, it was better than the one who was exhumed, deposed and burned as a heretic. Before the Gregorian Reform such accusations abounded against recently dead popes. As for ease of calculation appearing "to the medieval mind" as nothing short of diabolical, it has parallels in the way some modern treat me for speaking about subjects with assurance - because I have a certain sense of logic, of common sense, as lacking to them, as the positional system was to the contemporaries of Gerbert.
Due to his efforts to root out simony and other corruption within the Church, and his connection with science and intellectualism, there were many rumors and legends spread of Sylvester II being a sorcerer in league with the devil.
Source, wikipedia, Pope Sylvester II
Actually neither Wikipedia nor Catholic Encyclopedia online give anything like an indication his corpse was exhumed. One of the earlier Popes had been:
Shortly afterwards (4 April, 896) Formosus died. He was succeeded by Boniface VI, who reigned only fifteen days.
Under Stephen VI, the successor of Boniface, Emperor Lambert and Agiltrude recovered their authority in Rome at the beginning of 897, having renounced their claims to the greater part of Upper and Central Italy. Agiltrude being determined to wreak vengeance on her opponent even after his death, Stephen VI lent himself to the revolting scene of sitting in judgment on his predecessor, Formosus. At the synod convened for that purpose, he occupied the chair; the corpse, clad in papal vestments, was withdrawn from the sarcophagus and seated on a throne; close by stood a deacon to answer in its name, all the old charges formulated against Formosus under John VIII being revived. The decision was that the deceased had been unworthy of the pontificate, which he could not have validly received since he was bishop of another see. All his measures and acts were annulled, and all the orders conferred by him were declared invalid. The papal vestments were torn from his body; the three fingers which the dead pope had used in consecrations were severed from his right hand; the corpse was cast into a grave in the cemetery for strangers, to be removed after a few days and consigned to the Tiber. In 897 the second successor of Stephen had the body, which a monk had drawn from the Tiber, reinterred with full honours in St. Peter's. He furthermore annulled at a synod the decisions of the court of Stephen VI, and declared all orders conferred by Formosus valid. John IX confirmed these acts at two synods, of which the first was held at Rome and the other at Ravenna (898). On the other hand Sergius III (904-911) approved in a Roman synod the decisions of Stephen's synod against Formosus; all who had received orders from the latter were to be treated as lay persons, unless they sought reordination. Sergius and his party meted out severe treatment to the bishops consecrated by Formosus, who in turn had meanwhile conferred orders on many other clerics, a policy which gave rise to the greatest confusion. Against these decisions many books were written, which demonstrated the validity of the consecration of Formosus and of the orders conferred by him (see AUXILIUS).
The Source of that one is Catholic Encyclopedia : Pope Formosus./HGL