*A lecture room in the École Normale in post-Revolution Paris, in the year 1796. ENTER the Professor to give his lecture: PIERRE SIMON LAPLACE,*^{21} *aged 47.*

LAPLACE^{22}

Greetings, citizens! Today I shall discuss the notion of negative numbers, and the rules to be followed in operating with them. This is a topic which frequently stirs up anxiety and dissension - but the irresistible advance of Reason eventually triumphs over all petty rivalries and obstacles to the attainment of an end which is generally recognised as desirable, namely, the description and analysis of Nature in all her ways.

The properties and applications of negative numbers are now generally agreed. It will be one of your most important functions, when you return to the Provinces, to explain this (along with the new system of weights and measures which has just been definitively introduced by a decree of the National Convention) as victories for science and for the Revolution, to your fellow-citizens, and especially to the teachers!

Since there is nothing intrinsically difficult about negative numbers, children will find them much easier to grasp than many mathematicians! You will find it harder to explain them to teachers, who have long been familiar with the old restricted notion of number. They will find the new extended system of numbers difficult, because it is natural to think something is complicated when our prejudices and habits make it difficult for us to understand. However, I am confident that your intelligence and dedication will overcome these obstacles, and we may look forward to the day when these numbers are accepted and used everywhere.

[*He moves as if to stand looking up and down an East-West road, and uses grandiloquent gestures to illustrate the words East, West, Eastward and Westward, throughout the following* ]

Now, if numbers are to be useful for measuring not only absolute length, but also the *velocity* *v* of a carriage travelling Eastward on a straight East- West road, and the *distance* *s* of the carriage Eastward of a point on the road where we may suppose ourselves to be standing, then it becomes necessary to consider the orientation of those quantities: whether in fact the carriage is still *Westward* of us at a given moment, or whether its velocity is not in fact carrying it *Westward* instead of Eastward. And thus we find it most convenient to use the affectations or signs of the numbers +*v*, -*v*, and +*s*, -*s*, to denote the quantities taken in opposite directions East or West, respectively.

Supposing the time *t* of travel be measured from the moment the carriage passes us, then the later distance Eastward is given by the expression *s*=*vt*; but on the contrary, if the carriage be travelling Westward, the proper expression is -*s*=(-*v*)*t*; which is, however, just the original expression with the correct signs taken into account.

Now, supposing you should wish to know where the carriage was 10 seconds *before* it passed us, the unknown distance *s* is given by the expression *s*=*v*(-10) for the Eastward, and *s*=(-*v*)(-10) for the Westward journey. But travellers' commonsense tells us that the answers should be -10*v* and +10*v*, respectively, and so we have the rules: positive-by-negative is negative, and negative-by-negative is positive.

After all, it is not surprising that a reversal of a reversal should represent progress, or an enemy of an enemy should be a friend! As it is in life, so it is in Mathematics. Vive la Révolution! [CURTAIN *while students lustily echo: Vive la Révolution!* ]