June 9. 1910] 



NATURE 



423 



the remote epoch when it diverged from the Nilotic 

 culture, and Mr. and Mrs. Hawes's little book is de- 

 signed to instruct those who wish to know the story 

 of its origins. Felix qui potuit reriim cognoscere 

 causas. Religious ideas have largely directed the 

 general interest in our origins towards the " Bible- 

 lands," whence sprang the exotic oriental religious 

 element in our culture, but the growth of knowledge 

 and of civilisation is steadily weaning us from our 

 Semitic and mediaeval foster-parents, and interesting 

 us more and more in Greece and Rome, the real 

 parents of our minds and thoughts ; and the origjn 

 of Greece and of Rome was Crete, and Crete may 

 have sprung from the same common source as Egypt. 

 Of the Egyptian inspiration which we see in the 

 early art of Crete the authors of this little book say 

 little. They have no space in which to discuss dis- 

 puted points, and their personal bias is, perhaps, 

 rather away from any even so-called "oriental" in- 

 fluences (we do not admit, by the way, that Egypt 

 was ever " oriental " in the sense that the Semitic 

 world was and is). They merely describe what has 

 been found in Crete and is to be seen there, either in 

 the ruined palaces of Knossos and Phaistos, or in the 

 towns of Gournia and Palaikastro, or in the museum 

 of Candia, where the treasures found in the course of 

 the excavations of these places are preserved. They 

 conclude with a chapter on Cretan (Minoan) art which 

 strikes us as very correctly appreciative of the peculiar 

 genius of the earliest European artists, so unequal in 

 quality, so good, so magnificent in conception and 

 workmanship at times, at others so weak ; yet honest 

 and free, unshackled by any of the conventions that 

 bound the artists of Eg^pt and Assyria (who, but for 

 these conventions, would have done as well as the 

 Minoans), and the worthy ancestors and forerunners 

 of the artistic genius of Hellas. On this we must 

 always insist ; the Minoan art of Greece was 

 the ancestor of that of the Hellenes, who inherited 

 their artistic genius, not from the Indo-European 

 Greek-speaking northern originators of half their 

 blood, but from their other ancestors, the ruddy non- 

 .^an Mediterraneans, brothers of the Egyptian and 

 of the Etruscan. It is from these, albeit we ourselves 

 in the north have little or none of their blood in our 

 veins, that we have derived most of what makes us 

 civilised beings. 



PRACTICAL CURVE TRACING. 

 Practical Curve Tracing, with Chapters on Differentia, 

 tion and Integration. By R. Howard Duncan. 

 Pp. vii + 137. (London: Longmans, Green and 

 Co., 1910.) Price 55. net. 



THE methods employed in this book, which pre- 

 sents an attractive appearance, are almost en- 

 tirely independent of the aid of general mathematical 

 principles. For instance, the form of the graph of 

 y = ax + b and its dependence on a, b are explained 

 by plotting graphs of the equations obtained by vary- 

 ing o while h remains constant, and then those 

 obtained by varying b while a remains constant. 

 NO. 2 1 19, VOL. 83] 



Naturally greater difficulties occur in handling the 

 equations y = a.x- + bx + c, y=zax' + bx + c, &c., by the 

 same method. Inexpert mathematical students of the 

 tvpe for whom the author writes find it ver}' hard to 

 get hold of the notion of a parameter, and a great deal 

 could certainly be done by adopting the plan indicated 

 above, and steadily followed in this book. Even the 

 ordinan.' student of analytical geometry would prob- 

 ably get at " the facts of the case " sooner if he ap- 

 proached, for example, the equation x'-f y*— a.v— b = o 

 by drawing graphs of the circles of the specified 

 system, keeping b a positive constant and giving a 

 various values, then keeping b a negative constant 

 and varying a. 



It is this positive and distinct advantage that is 

 emphasised by the author, and from this point of view 

 are discussed the parabolic, hyperbolic, exponential, 

 and logarithmic cun'es, together with the sine curve, 

 of the natures of which a good account is given. For 

 students of graphs who have at their disposal 

 algebraic machiner\^ up to division and quadratic 

 equations, the road to a knowledge of the forms of 

 many graphs could be made shorter. The artifices of 

 change of origin and scale-unit, even without those 

 of successive approximation, do not offer a great 

 difficulty to a student of small mathematical ability, 

 and go a long way towards establishing the rough 

 form of the graph of an equation which would appear 

 alarming if it had to be discussed by the plotting of 

 points. 



Two chapters on the calculus are added to those on 

 cur\-e tracing. The author knows that " the method 

 of measuring the slope of a curve by actually draw- 

 ing the tangent is sometimes objected to on the 

 ground of inaccuracy " ; but his experience " shov.s 

 that by good and careful workmanship it is possible 

 to rely on the results so obtained to a degree of 

 accuracy which is sufficient for practical purposes." 

 Yet the degree of accuracy indicated in some of the 

 results tabulated in the chapter on differentiation must 

 be very difficult to attain. Indeed, cur\-es of y = .v-, 

 y = .\'', &c., are constructed, tangents are actually 

 drawn, dyjdx and x are tabulated and then plotted 

 against one another directly or logarithmically, with 

 so much accuracy that the rules for the differentiation 

 of .r„, e', log -v, sin x, cos .v, are deduced. The reader 

 certainly will have it ven.- definitely impressed upon 

 him that dyidx measures the slope of a cur\-e. Of 

 course, there remains the difficulty for an engineer, or 

 any other who applies the calculus, of being able to 

 identify the slope with the rate of variation of the 

 corresponding function, and of appreciating the very 

 varied significance of the derivative in its applica- 

 tions ; but the book does not profess to enter on this 

 field. 



A few examples on each chapter are gathered to- 

 gether at the end of the volume, the purpose of which 

 is evidently that the reader should be clear regarding 

 the facts at the base of the equations and functions 

 discussed before he sets out to equip himself in the 

 practice and applications of the methods explained. 



P. P. 



