Here we have two words, one in Arabic and one in Hebrew, which scholars of those languages will have no difficulty in recognizing as descending from the same ancient Semitic source. In Arabic the word means “error” in the sense of “error message” from a computer. The Arabic version is pronounced “khata” (with an emphatic “t”), and the Hebrew is quite similar.

I’m sticking with the Arabic for now, because this is the jumping-off point for an interesting bit of medieval mathematical history.

The word “algebra” comes to us from the seminal work of Al-Khwarizmi (c.780 – c.850), who first gave the world a

**procedure for solving quadratic equations. He lived in Baghdad, but the Egyptian Abu Kamil Shuja (c.850 – c.930) is also of importance, not only for “upgrading” his work but also transmitting it further west where it was picked up by the great Leonardo da Pisa, aka Fibonacci.**

*systematic*One particular procedure that was codified by Abu Kamil was a method of solving simultaneous equations. A method of solving simple equations, known as the Method of False Position, had been known (with variations) to the Ancient Egyptians and Chinese. But Abu Kamil gave the world the

**Method of Double False Position**, which dealt with simultaneous equations. In Arabic this is

**حساب الخطأين**or

**Hisab al-Khata’ein**(see our word in the title.) If he did not invent it himself, he was at least the first to publish a systematic treatise on its use.

This method was, perhaps, a bit advanced for most medieval applications, but in Tudor times it received a great boost from Robert Recorde (1510 – 1558), the Welshman who invented the “=” sign. Here is the method in his work

*The Ground of Artes*, there called “

**A Discussion of Sheep**”

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

How many sheep may be kept on a 7000-acre farm if law requires that there be one acre of arable land for every 10 sheep, and one acre of pasture for every 4 sheep?

The student makes guesses of 500 and 1000 sheep, and writing these numbers down, calculates that to keep this many sheep a farmer would need only 175 and 350 acres respectively.

Noting that these figures are ‘out’ by some 6825 and 6650 acres . . . . . the student produces the following diagram.

And next he performs the calculation:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

It’s a remarkable bit of work, but explaining it algebraically is somewhat tedious. But where does the Hebrew come in? Well, here it is from my favourite website for exploring the Tanach:

Recognize our word at the bottom? The English translation goes:

1 [A Psalm] of David. Maschil. Happy is he whose transgression is forgiven, whose sin is pardoned.

My guess is that the ancient Semitic meaning is “to miss the target”: in the Hebrew (getting on for 3000 years old) I doubt if they differentiated much between the meaning in hunting and the more ethical (or whatever) meaning of “sin”. The Arabic we know today is from about half that time ago, and especially with the development of philosophy in old Baghdad the technical and religious meanings would become quite distinct: for technical use the original word is retained, and for “sin” the derived word “khati’a” is used instead.

But unlike the mathematical method, I don’t think that in this case two wrongs make a right!

= = = = = = = = = = = = = = = = = = = = = = =

Here are some interesting references to this method:

[1] The link “Robert Record” on http://myweb.tiscali.co.uk/annotations/history_of_maths.html

[2] http://www.codewit.com/castleofknowledge.php

[3] and one with y

^{e}original Tudor English: http://sflip.excellencegateway.org.uk/pdf/Numeracy%204.9%20Session%201%202006.pdf

whiche I gladlye reproduce below:

**A discussion of sheep**

SCHOLAR: There is supposed a lawe made that (for furtheryng of tyllage) every man that doth kepe shepe, shall for every 10 shepe eare and sowe one acre of grounde, and for his allowance in sheepe pasture there is appointed for every 4 shepe one acre of pasture. Nowe is there a ryche shepemaister whyche hath 7000 akers of grounde, and woulde gladlye kepe as manye sheepe as he myght by that statute. I demaunde howe many shepe shall he kepe?

MASTER: Answere to the question your selfe.

SCHOLAR: Fyrste I suppose he maye kepe 500 shepe, and for them he shall have in pasture, after the rate of 4 shepe to an acre, 125 akers, and in earable grounde 50 acres that is 175 in all, but this errour is to litell by 6825. Therefore I gesse agayn that he maye kepe 1000 shepe, that is in pasture 250 akers, and in tyllage 100 akers, which maketh 350, that is to lytle by 6650. These bothe erroures with theyr positions I sette downe as you see, and multiply in crosse 6825 by 1000, and it maketh 6825000. Then I multiply 6650 by 500, and it doth amount to 3325000, which sum I do subtract out of the fyrst,&there remaineth 3500000, as the dividende. Also I doo subtract the lesser errour out of the greater, and so remayneth 175, by which I divide the said dividende, and the quotient will be 20000, so that I see that by this rate he that hathe 7000 acres of ground may keepe 20000 shepe:&therby I conjecture that many menne may kepe so many shepe: for many men (as the common talke is) have so many acres of grounde.

MASTER: That talk is not likely, for so muche grounde is in compas above 48¾ miles. But leave this talke&returne to your questions, least your pointing be scarse wel taken.

SCHOLAR: In dede I doo remembre that the Egyptians did grudg so much against shepards, till at length thei smarted for it,&yet they were but smal shepemaisters to some men that be now, and the shepe are waxen so fierce nowe and so myghtye, that none can withstande them but the lyon.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Note: “scarse wel taken” – there was a political edge to this, what with changes of land use at the time.

## Comments