66 PHILOSOPHICAL TRANSACTIONS. [aNNO 1684. 



antecedents a and c are either equal to their consequents or greater or less; if 

 equal the thing is manifest ; if less, then by the definition of proportionals a 

 and c are like parts of b and e, therefore what proportions the whole b and e 

 have to one another, the same will a and c have, which is permutation ; like- 

 wise e:c::b:a, which is inversion. So also, if from proportionals we take 

 like parts, the remainders will be proportional; whence conversion and division 

 are demonstrated ; and if to proportionals we add like parts, the wholes will 

 still be proportional, which is composition, &c. If the antecedents be greater 

 than the consequents, the consequents will be like parts of them, and the de- 

 monstration exactly the same with the former. 



The first of the 6th book is proved by considering the generation of parallel- 

 ograms, which are produced by drawing or multiplying the perpendicular upon 

 the base, that is, taking it so often as there are parts and divisions in the base; 

 therefore, in fig. 6, the same proportion that RX single has to NX single, the 

 same has RX multiplied by XZ, that is, repeated a certain number of times, to 

 NX multiplied to XZ, that is, repeated the same number of times, which is as 

 much as to say, RX : NX :: the paral, RZ : the paral. NZ; now that this pro- 

 portion also is true in oblique angled parallelograms is proved, because they are 

 equal to rectangled ones on the same base and between the same parallels, as 

 thus independently appears; the triangles RQX and MPZ are equal, for RX 

 = MZ, QX = PZ, RM = QP, therefore adding MQ to both, RQ = MP; 

 if therefore from these equal triangles we take what is common, viz. MLQ, 

 the remainders will be equal, viz. RXLM = QLZP; to both of those add 

 XLZ, then the whole parallelograms will be equal, viz. RZ=QZ, Q. E. D. 

 That triangles also having a common base are in the proportion of their altitudes, 

 hence follows, because they are the halves of parallelograms on the same base. 

 And the demonstration is exactly the same in prisms, pyramids, cylinders, and 

 cones, having the same base. 



To prove the l6th of the 6th book, I suppose the 4 lines a, b, c, e, to be 

 proportional ; that is, granting a and c to be the less terms, the same way that 

 a is contained in 6 so is c in e, and that d is the denominator of the ratio, it will 

 follow then that b is made up of a multiplied by d, and e of c multiplied by d, 

 so that ad = b and cd=:e, draw therefore the extremes upon one another, that 

 is a on cd, and the means, that is, c on ad, the factors being the same, the 

 products acd and cad are the same, and consequently equal Q. E. D. 



An Account of the Course of the Tides at Tonquin; in a Letter from Mr. Francis 



Davenport. N° l62, p. 677. 



The peculiar circumstances of the tides at Tonquin are sufficiently stated in 



