October 1, 1890.] 



KNOWLEDGE 



229 



of the actual facts, his book would have merited almost 

 imreserved commendation. 



It may be added that his readinp:, thonj^h extensive, has" 

 by no means been exhaustive. Among tlie more im- 

 portant sources of information which he has overlooked 

 may be enumerated the Rectitudines Simiuhirum I'eraona- 

 ruiii, a most valuable Anglo-Saxon treatise on village 

 customs and the rights and duties of ofllcials. To this 

 treatise, which should have been his text-book, he only 

 once refers, and that at second-hand. The evidence of 

 the Boldon Book, as compared with Bishop Hatfield's 

 subsequent survey of the same baronies, as well as the 

 Black Book of Hexham, would have supplied valuable 

 evidence as to early tenures, and the gradual changes 

 which took place in them with the lapse of time. The 

 volume of Dmii/'fidiii/ Sti(dii'x, which was the fruit of the 

 Domesday Commemoration of 1886, with which he does 

 not seem to be acquainted, would also have elucidated 

 some obscure points relating to communal tillage, land 

 measures, and taxation. 



NOTES ON PHYLLOTAXY, OR THE MATHE- 

 MATICAL ARRANGEMENT OF LEAVES 

 AND BRANCHES. 



By .T. Pbntland Smith, M.A., B.Sc. 



Tin^ flowering plant shown in the plate Saxifnif/a 

 IJi/i-i{iiiidiilis has become very popular of recent 

 years, by reason of its fine trasses of flowers and 

 hardy character which render it suitable for house 

 decoration. As its generic name would imply, it 

 belongs to the family, or Natural Order, Saxifi-ageiie. The 

 plants whieli make up this Natural Order all possess 

 two seed-leaves or Cotyledons, and are inchided in that 

 large group of plants known as Dicotyledons. Further- 

 more, as the stamens, or male organs, and the petals, or 

 coloured portions, of the flower, appear to arise from the 

 modified green leaves, the sepals, occupying the peri- 

 phery of tlie flower, the Saxifrages are classed under that 

 division of the Dicotyledons termed the Calyciflone. The 

 plants belonging to the Order of SaxifragefB, are allied to 

 those forming the Leguminosa? (Pea family), Rosaceaj (Rose 

 family), and Crassulacea^ (Stone-crop family). In the Rose 

 there are glands at the base of the pistil or female part of 

 the flower, which are absent in the Saxifrages. In this 

 Order are found the Australian pitcher-plant, and as first 

 cousins of the specimen before us we may rank the fami- 

 liar (iooseberry and the Red and Black Currants. 



It is rather curious to find that the Saxifrages obtained 

 their name from a remarkable property which they were 

 supposed to possess, viz. that of breaking stones {sii.i-um, 

 a rock, and frani/o, to break) in the bladder. 



The arrangement of the leaves and flower-bearing 

 brandies shown in the picture may at first ap]iear to be 

 quite fortuitous, but a closer inspection will satisfy the 

 student tliat a definite law governs the arrangement, not 

 only of the leaves around the foot, but also of the flower- 

 bearing branches which spring from leaves at their base 

 or junction with the central stem or axis of the pyramid of 

 flowers. 



A plant derives in part its characteristic appearance 

 from tlie manner in which the leaves arise from the stem. 

 Tlie arrangement of the leaves on tlie stem is termed 

 jiliilllotd.ris ((ireek (jivXXov, a leaf, and Tu$t.<;, arrangement). 

 At iirst sight it appears that the leaves are scattered on 

 the stem without any regard to order or regularity ; but by 

 an attentive examination it can be seen that here, as in 



other departments of Nature, the reign of law can every- 

 where be traced, and that a stem always sends forth its 

 leaves subject to mathematical laws. The an-angement of 

 the leaves and branches varies with iliff"erent plants ; some- 

 times tlie phyllotaxis even changes on dift'erent parts of the 

 same plant, and sometimes it remains constant through- 

 out a whole genus, and occasionally a certain phyllotaxis 

 is characteristic of a whole order or natural group of 

 plants — for example, in the LnhinUi-, the Dead Nettle 

 (Jrder, the leaves are always placed opposite to one 

 another. 



Besides the opposite arrangement just cited, there is 

 the irliorled arrangement, in which three or more leaves 

 apjiear at the same level on the stem, and the xnittered, or 

 alt/'nidfi', arrangement, in which each leaf arises at some 

 distance from its neighbours. The points at which the 

 leaves arise are termed the nodes, and the intervals be- 

 tween them are termed internodes. 



The internodes may be extremely short, as for example 

 in I'lindanns, the Screw Pine, in the House Leek, and the 

 lower portion of the stem of the Saxifrage of om- illustra- 

 tion. There it can be seen that the leaves form a spiral 

 or ludix, which winds roimd the axis on which they arise. 

 This is particularly visible in the Pandanus, as anyone 

 who has visited the Palm House at Kew must have 

 noticed, and in this case the sphal 

 seeil is what is termed the ijenetic 

 sphal. It is so called because it 

 includes in it every leaf as it de- 

 velops. It is the spiral of growth. 

 In stems in which the alternate 

 arrangement of leaves prevails, 

 such as those of our common forest- 

 trees, there are great diflerences 

 both as regards the length of the 

 internodes and the number of 

 leaves which have to be passed 

 over, in tracing out the genetic 

 spiral, before a leaf is arrived at 

 which is disposed vertically above 

 the one with which the observer 

 has started. Take a t-nig of the 

 Elm or Beech, select a leaf, then 

 p J trace out a spiral which will in- 



clude every leaf. Calling the first 

 leaf 0, and nnnibermg the successive leaves 1, 2, 8, \, and 

 so on, you will find that leaf No. 2 is superposed to leaf 

 No. 0, and that the spiral you have described lias wound 

 once round the stem (see Fig. I. («)) ; thus the dis- 

 tance, or anijuliir diverijnice between any two leaves in 

 the spiral is one-half the circumference of the stem, or 

 180°. 



Had a branch of the Elder or Aspen been selected, in 

 describing one turn round the stem leaf No. 3 would 

 have been found vertically above leaf No. (see Fig. I. (/<)), 

 and the anguhir divergence would have been 120°. Two 

 turns of the spiral must be made in the case of the young 

 shoots of the Oak or Willow before two leaves are foimd 

 iu a vertical line, and then -5 stands above 0, thus making 

 the angular divergence 15i°, and the spiral a | arrange- 

 ment (see Fig. I. (c)). 



In the Holly, leaf No. 8 stands vertically above leaf 

 No. 0, and the spiral has passed three times aroiuid the cir- 

 cumference of the stem ; hence the angular divergence is | 

 of 360°, or 135° (see Fig. I. (./)). An angular interval of ^ 

 is seen in the Lily and in the cones of Pinus ulndnis (the 

 Weymouth Pine), and of /j in the cones of many other 

 Pines. Placing these fractions together, we get a series — 

 a. ^i l> fi 2\> etc., which can easily be remembered, as the 



