The remainder of this chapter may be omitted by readers who have not even the most elementary acquaintance with geometry or algebra. But for the benefit of those whose education has not been _entirely_ neglected, I will add a few explanations of the general formula of which I have hitherto given only particular examples. The general formula in question is the “Lorentz transformation,” which tells, when one body is moving in a given manner relatively to another, how to infer the measures of lengths and times appropriate to the one body from those appropriate to the other. Before giving the algebraical formulæ, I will give a geometrical construction. As before, we will suppose that there are two observers, whom we will call =O= and =O=′, one of whom is stationary on the earth while the other is traveling at a uniform speed along a straight railway. At the beginning of the time considered, the two observers were at the same point of the railway, but now they are separated by a certain distance. A flash of lightning strikes a point =X= on the railway, and =O= judges that at the moment when the flash takes place the observer in the train has reached the point =O=′. The problem is: how far will =O=′ judge that he is from the flash, and how long after the beginning of the journey (when he was at =O=) will he judge that the flash took place? We are supposed to know =O=′s estimates, and we want to calculate those of =O=′. [Illustration] In the time that, according to =O=, has elapsed since the beginning of the journey, let =OC= be the distance that light would have traveled along the railway. Describe a circle about =O=, with =OC= as radius, and through =O′= draw a perpendicular to the railway, meeting the circle in =D=. On =OD= take a point =Y= such that =OY= is equal to =OX= (=X= is the point of the railway where the lightning strikes). Draw =YM= perpendicular to the railway, and =OS= perpendicular to =OD=. Let =YM= and =OS= meet in =S=. Also let =DO′= produced and =OS= produced meet in =R=. Through =X= and =C= draw perpendiculars to the railway meeting =OS= produced in =Q= and =Z= respectively. Then =RQ= (as measured by =O=) is the distance at which =O′= will believe himself to be from the flash, not =O′X= as it would be according to the old view. And whereas =O= thinks that, in the time from the beginning of the journey to the flash, light would travel a distance =OC=, =O′= thinks that the time elapsed is that required for light to travel the distance =SZ= (as measured by =O=). The interval as measured by =O= is got by subtracting the square on =OX= from the square on =OC=; the interval as measured by =O′= is got by subtracting the square on =RQ= from the square on =SZ=. A little very elementary geometry shows that these are equal.