sausage-rolls, and 6 bottles of Zoëdone." (Poor souls, how thirsty they must have been!) If Balbus will kindly try this by his principle of "two assumptions," first assuming that a bun is 1 d. and a queen-cake 2 d. , and then that a bun is 3 d. and a queen-cake 3 d. , he will bring out the other two luncheons, on each assumption, as "one-and-nine-pence" and "four-and-ten-pence" respectively, which harmony of results, he will say, "shows that the answers are correct." And yet, as a matter of fact, the buns were 2 d. each, the queen-cakes 3 d. , the sausage-rolls 6 d. , and the Zoëdone 2 d. a bottle: so that Clara's third luncheon had cost one-and-sevenpence, and her thirsty friends had spent four-and-fourpence!
Another remark of Balbus I will quote and discuss: for I think that it also may yield a moral for some of my readers. He says "it is the same thing in substance whether in solving this problem we use words and call it Arithmetic, or use letters and signs and call it Algebra." Now this does not appear to me a correct description of the two methods: the Arithmetical method is that of "synthesis" only; it goes from one known fact to another, till it reaches its goal: whereas the Algebraical method is that of "analysis": it begins with the goal, symbolically represented, and so goes backwards, dragging its veiled victim with it, till it has reached the full daylight of known facts, in which it can tear off the veil and say "I know you!"
Take an illustration. Your house has been broken into and robbed, and you appeal to the policeman who was on duty