work in the book which he wrote more than seven centuries ago. He first wrote the multiplicand; above that he placed the multiplier; and above that he wrote the product. This is our plan turned upside down. If you could ask Leonardo why he turned the work
HOW LEONARDO MULTIPLIED
From Leonardo's arithmetic of 1202. It representsthe multiplication of 49 by 8, the product being 392
upside down, he would say that his work was right, but ours is upside down. It all depends on how we are taught.
It was about three hundred years after Leonardo that Cuthbert Tonstall (tun'st l) was born in England. In those days it was still the custom in most countries to compute by counters, but Cuthbert went to a school where they taught the new method. In 1522 he published the first arithmetic printed in England. Although this was about a hundred
OF LONG AGO
years before the Pilgrims landed at Plymouth, it is interesting to know that only thirty-four years later, in 1556, an arithmetic was printed in the City of Mexico.
We do not know how Cuthbert was taught to multiply, but his book tells us what he thought was the best method. This was very
HOW CUTHBERT MULTIPLIED
From Cuthbert Tonstall's arithmetic of 1522, showing the method ofmultiplication used by him
much like our own except that he did not know our short way of multiplying by zero, or, as he called it, " by a circle."
Thus we see that this great man could not multiply as easily as you can.
Just before Cuthbert's time there was a boy named Johann Widman (y hän' v d'män)
PAGE FROM JOHANN WIDMAN'S BOOK
This shows two multiplication tables from Johann's arithmetic country to teach arithmetic in all schools. Those who were going into business went to an arithmetic school and learned from an 68
OF LONG AGO
arithmetic teacher. In this school Johann learned how to multiply in the way that Cuthbert did, although he also learned how to multiply with counters as the German merchants did at that time.
When Johann became a man he wrote an arithmetic, and in this he used the signs for plus ( +) and minus ( ). This was the first time these signs were ever printed.
The way in which Cuthbert and Johann multiplied was not the only one followed in those days. Some teachers ruled the paper in squares like a chessboard, writing a figure in each square, and they called their plan the chessboard method. Others ruled the paper so that it looked like an iron grating, and they called this the grating method.
These are some of the ways in which the world tried its hand at multiplying, and that is all the story we shall have to-night about Cuthbert and Leonardo and Johann.
" I don't see," said the Tease, " how they ruled the paper in the grating plan. I don't see what it means."