687 



DUPLICATE RATIO. 



DUST, ATMOSPHERIC ; DUST-HAZE ; DUST-STORM. 098 





from the archbishop to the crown in council. It seema to have been 

 called double querele, because in its form it is a complaint both against 

 the judge and against the party at whose suit justice is delayed. 

 (Blackst. ' Comm.,' Mr. Kerr's edit., iii. 256.) 



DUPLICATE RATIO (\6yos Snr\atrlav), a term used by Euclid, and 

 denned aa follows : If A be to B in the same proportion as B to c, then 

 the ratio of A to c is called the duplicate ratio of A to B. When A, B, 

 and c are lines, the duplicate ratio of A to B is that of the square on A 

 to the square on B ; when numbers, that of A times A to B times B. 

 [RATIO ; EXPONENT.] 



DUPLICATION OF THE CUBE, the solution of the following 

 problem : to find the side of a cube which shall be double that of 

 another cube. This question, which is insoluble with perfect exact- 

 ness by the methods of ordinary geometry, attained such a degree of 

 notoriety among the Greek geometers that its origin was the subject of 

 a mythologic fable. Eutocius, in his commentary on the sphere and 

 cylinder of Archimedes, has preserved a letter of Eratosthenes to Ptolemy 

 (Euergetes) in which it is said that one of the tragedians (Euripides, 

 according to Valckenaer, cited by Montucla's editor) had introduced 

 Minos erecting a sepulchre to Glaucus. The architect proposed one 

 hundred palms every way, on which Minos declared that such a size 

 would be too small for a royal sepulchre, and required that it should 

 be doubled in size ; and thereupon arose the difficulty. Eratosthenes 

 also states another fable, namely, that the Delians, during a pestilence, 

 had been ordered by the oracle to produce a cubical altar double of one 

 which then existed. They applied to the school of Plato at Athens, 

 who found that the, problem eluded all their efforts. Other writers 

 make mention of the latter story, and Valerius Maximus. in particular, 

 adds that Plato referred the querists to Euclid ; which must be an 

 anachronism. However this may be, the problem continued to furnish 

 an unceasing object of research ; and such was the importance of its 

 solution in the eyes of Eratosthenes, that he hung up his own solution 

 in a temple as an offering, and composed an epigram, of which the 

 principal value now is the proof which it affords that he considered 

 Memechmus as the first inventor of the conic sections. 



Hippocrates of Chios, known as the first who could find the area of 

 a (not any) curvilinear figure, perceived, according to Eratosthenes, 

 that this problem could be solved as soon as two mean proportionals 

 could be found between the side of the given cube and twice its 

 length ; that is, A being the length of the given cube, and x and y two 

 lines such that 



A : x : : x : y and x : Y : : Y : -2 A, 



this geometer saw that x was the side of the cube double of that on A. 

 But the new problem presented exactly the same difficulty as before ; 

 various mechanical curves (as they were called) were invented for the 

 purpose ; it was found that the conic sections were sufficient, but no 

 solution appeared consistent with the restrictions implied in the postu- 

 lates of Euclid. 



Eutocius has mentioned the solution of Eudoxus, and has preserved 

 those of Plato, Hero, Philo, Apollonius, Diocles, Pappus, Sporus, 

 Menicchmus, Archytas, Eratosthenes, and Nicomedes. Pappus himself 

 (in the third book, the first of those which remain entire) has preserved 

 the solutions of Eratosthenes, Nicomedes, and Hero. In several 

 instances these notices are the only clue which we have to the dates of 

 the investigators, as there is strong presumption that those who arc 

 named by Eutocius and not by Pappus lived between the two. 



The trisection of the angle [TRISECTION] offered difficulties of a 

 similar kind, and engaged the attention of several of the individuals 

 above mentioned. That of the quadrature of the circle is altogether of 

 another kind. For the various solutions of the problem of the dupli- 

 r.iti'iii, see Montucla, ' Histoire des Recherches sur la Quadrature du 

 i 1 edition, Paris, 1831 ; or Reimer, ' Historia Problematis de 



C'ubi Duplicatione,' Gottingen, 1798; or the works of Eutocius and 

 Pappus already cited. 



The importance of this problem declined with the rise of the decimal 

 arithmetic. Many difl'erent attempts were made, some avowedly 

 mechanical (as opposed to geometrical), others by those who imagined 

 they could overcome the original difficulty. Any process for the solu- 

 tion was calif, d mcsolabum (a term as old as Vitruvius). One of the 

 last was that of the celebrated Vieta, containing an error, which is the 

 more remarkable, that little, if any, notice has ever been taken of it. 

 (See his works, Schooten's edition, p. 273.) 



IH'ST, ATMOSPHERIC; DUST-HAZE; DUST-STORM; are 

 terms designating related phenomena, depending on the suspension in 

 the air of congeries of solid particles, which, within about a quarter of 

 a century past have received some definite scientific attention ; finely- 

 divided solid matter, derived from the mineral kingdom, together with 

 minute organisms or fragments of animal and vegetable origin, having 

 'o be regarded as forming a proximate element of the lower strata 

 of the atmosphere, regarded in the gross. Their suspension, no doubt, 

 is dependent on the principle of the internal friction of the air, to 

 which Professor Stokes has referred that of the clouds. [CLOUD.] 



Thc*grcat pioneer of systematic and scientific micrology, Professor 

 iberg of Berlin, who has done for a host of the objects of natural 

 history n- 'jtiiriug the microscope for their investigation, or even dis- 

 tinct vision, nearly what Linnaeus did for all the species of organic 

 nature known to ham, which could be examined by the unassisted, or 



but slightly assisted, eye, has not omitted to extend his philosophic 

 scrutiny and distribution to the motes that float in the air, referring 

 them to their places in nature, or to the creatures from which they 

 have been derived. In 1849, he published collectively the results of 

 the researches on this subject which he had then been pursuing for 

 some years, under the title of ' Dust of the Regular- Winds and Blood- 

 rains,' &c., consisting of 192 folio pages, and containing seven tabular 

 catalogues, and six coloured plates of particles of atmospheric dust as 

 viewed by the microscope. A synopsis of the contents of this work 

 will be found in the ' Pharmaceutical Journal,' edited by the late Jacob 

 Bell, for June 1850, vol. ix., pp. 569-575. From this it appears that 

 the dust of the regular winds yields to chemical examination silica, 

 alumina, potash, soda, oxide of iron, oxide of manganese, oxide of 

 copper, carbonate of lime, magnesia, water, and organic (combustible) 

 bodies. By microscopic analysis are discerned in it fine quartz sand, 

 still finer yellowish or reddish mould (very fine granular dust, Galliu- 

 nella ferruginea ?), mixed with numerous organic forms and fragments. 

 Generally single isolated fragments of pumice are observed in it, but 

 chiefly green prismatic crystals, as in volcanic tufa and rapilli. Like- 

 wise white calcareous crystals which readily dissolve in muriatic acid. 

 The organic substances are Pobjrjastrica, Phytolitharia, Polythalamia, 

 and soft vegetable parts. The total number of organic forms observed 

 at this period amounted to 320 species ; but many more have since 

 been discovered, and have been described in Professor Ehrenberg's 

 communications to the Royal Academy of Sciences at Berlin, con- 

 tinuing the elaborate investigation of which we here present merely a 

 few results. 



Over the island of Teneriffe, as recorded in Professor C. Piazzi 

 Smyth's Report on the astronomical experiment in that locality, in 

 the year 1856 the principal object of which was to ascertain, by 

 following out a suggestion of Newton, how much celestial observation 

 can be benefited by eliminating the lower third or fourth parts of the 

 atmosphere the existence of atmospheric dust is manifested in a 

 remarkble manner, affecting all the phenomena dependent on the 

 transparency and the transcalescence of the air. When describing the 

 astronomical qualities of the atmosphere at his observing stations ou 

 the Peak, and after noticing those arising from the winds and the 

 clouds, he thus describes those which are imparted by the dust : " A 

 more important quality of the atmosphere was caused by the dust-haze, 

 which was ever more or less present, though sometimes in vastly greater 

 quantities than at others, and was precisely that which injured, or 

 rendered impossible, daylight observations of stars (weakening direct 

 light and multiplying general light). Where this dust-haze came 

 from or went we could never tell ; but, when present, we could easily 

 distinguish its banks, or strata dense strata, far above the clouds of 

 the north-east trade-wind as they stretched away and condensed in 

 perspective towards the horizon. There were often several strata, one 

 above the other, and mutually separated by very clear and sharply- 

 defined spaces of atmosphere. When, as was sometimes the case, the 

 summits of Grand Canary, or of Palma [other islands of the group], 

 rising high above the sea of clouds, pierced also these upper strata of 

 dust-haze, wu had from [the observing station of] Guajara, the curious 

 phenomenon of zones of blue mountain alternately distinct and again 

 indistinct almost to invisibility, and yet no cloud or other recognised 

 impurity of the atmosphere intervened. Being above much of this 

 dust, though perhaps not the greater part of it [the elevation of the 

 station being 8903 feet, and the dust-strata rising to not less than 

 11,000 feet], we were evidently better off than an observer at the level 

 of the sea, when pointing to a zenith object, but for a horizontal one 

 we were worse off, from often being in, and then looking through the 

 whole plane of the stratum, and so experiencing the maximum of its 

 light-stopping effect. Hence the occasional deterioration of sunrise 

 and sunset were infinitely greater than anything that occurred at noon ; 

 and on some days, when the sky was perfectly free from cloud, and the 

 sun had been distressingly hot and bright when high in the sky, yet 

 it had almost become invisible before it set. It was seen, though 

 made out with difficulty on such occasions, through a darkling, yet 

 luminous haze of dull lemon-yellow colour ; but what it set behind, or 

 when exactly it did set, there was no ascertaining. The next evening 

 perhaps the atmospheric dust had removed, and the change in the 

 sunset was magical. The orb radiated hot and bright up to the 

 moment of going down. Then, too, in place of the uniform yellow 

 colour of the dusty sunset, the most gorgeous scarlets, yellows, and 

 blues took its place." 



From his experiments on the radiation of the sun, Professor Smyth 

 says on this subject, after drawing some important inferences, 

 " Towards the chief astronomical end of the expedition, there is a yet 

 more interesting conclusion to be drawn. The days of highest radia- 

 tion are those of least temperature, and m'c versA : and this difference 

 obtains in a signal degree when there was no visible disturbing action 

 of wind or clouds. What then causes the radiation of one day to be 

 greater than that of another, and the temperature loss ? The imme- 

 diate agent appears to be the atmospheric dust," diminishing by its 

 interposition the effect of the solar radiation, but at the same time 

 augmenting the temperature of the air by the multiplied radiation and 

 reflection from its particles, in a manner corresponding to its action 

 upon light as already noticed. " Hence then, we may easily under- 

 stand why the small difference of altitude [1800 feet] between Alta 



