Если куб тремя взаимно перпендикулярными центральными плоскостями разделим на восемь кубиков, то получим периодическую операцию, которую бесконечно можем продолжать в обе стороны. При этом объем куба уменьшается (или увеличивается) в восемь раз, то есть два в кубе. Но этот период очень просто разделяется на три меньшие периода с уменьшением (или увеличением) объема в два раза. В этой периодичностн особенную роль играют А) центры граней исходного куба и В) центры кубов низшего периода.
В статье "Результаты первой стадии экспериментального исследования структуры кристаллов» основным параллелоэдром обозначена тот наименьший параллелоэдр с отмеченными в нем расположениями атомов, из которого посредством элементов симметрии связи выводится вся система атомов. В системе атомов имеется поэтому несколько различных ориентировок основных параллелоэдров с их атомами, а именно по величине симметрии связи (см. статью).
Прямой опыт показывает, что явление кристаллизации есть явление весьма сложное, в котором решающую роль играют различные факторы. Хотя по закону Стено углы между соответственными гранями в кристаллах одного и того же вещества постоянны, но появление форм, обусловливающнх кристаллизацию, далеко не отличается полным постоянством. С первого раза представляется даже нечто обратное. От одного и того же вещества можно иметь кристаллы с очень ограниченным числом пар граней, в исключительных случаях не достигающем даже трех, а можно иметь и кристаллы с очень богатой комбинацией. Формы, слагающие комбинация, также весьма разнообразятся по своему развитию, и те самые формы, которые в одном случае являются преобладающими по своей величине, в других случаях отходят на второй план или даже совершенно подавляются другими формами.
Для установления структуры мы и раньше руководствовались не полною суммою комбинаций, какие проявляет данный кристалл, а только небольшим числом важнейших пар граней. В этом отношении, как видим, никакого изменения не произошло. Мы и теперь должны прежде всего отметить эти важнейшие формы, но раньше мы делали расчет плотности этих форм, так как все-таки в основе лежала последовательность граней по важности. Теперь этот расчет отпадает и даже значительно упрощается, потому что остается только по этим формам определить, в какой из трех поясов попадают эти формы.
Первоначальным материалом послужил горючий сланец, собранный при разведочных работах Н. Ф. Погребова. После отгонки светильнаго газа остается смола, послужившая материалом для данного исследования. При перегонке этого материала в водяной бане в трубке садятся кристаллические, весьма тонкие пластинки перегоняющегося органического вещества со включениями другого вещества с значительно большим преломлением и по-видимому способного к очень хорошей кристаллизации. Эта перегонка производилась г. Вальчисом.
Одна теорема состоит в том, что расстояние гномостереографической от линейной проекции в некоторой плоскости равно расстоянию от последней точки схода лучей. Доказательство сводится к тому, что точка схода лучей Z, гномостереографическая проекция Р и средняя точка линейной проекции плоскости О составляют вершины равнобедренного трехугольника, имеющего в основании первые точки, а это, в свою очередь, сводится к доказательству равенства углов при основании.
Так как на плоскости аналогичная задача разрешается весьма просто на основашн теоремы Паскаля, то весьма естественно, что мысль геометров упорно направлялась на отыскание простого решения вопроса для пространства, а неудача подняла этот вопрос на степень трудной проблемы. Теорему Паскаля можно формулировать различным образом, но, особенно с точки зрения современной reoметрии, для которой она послужила одной из первых основ, эту формулировку приходится связывать с коллинеацией, а именно такою, которая преобразует определяемую пятью точками коноприму, в самое себя. IIocтpoeние по теореме, аналогичной Паскалевской, переносится во все геометрические системы, в частности, и в систему плоскостей, но последняя теорема, как теорема непозиционного характера переносится только на родственные системы, а например, на систему плоскостей не переносится, также как для системы лучей на плоскости не применима только что приведенная теорема непозиционного характера.
Обратимся к аналогичным построениям в пространстве, являющимся следствием теоремы, аналогичной теореме Паскаля. Так как в основании построения по этой теореме находится nocтpoeние двух гиперболоидов линейной примы, к которой принадлежит и искомая коносекупда, а для этого нужно построить две гексапримы, то ясно, что данными могут являться такие касательные совокупно с точками касания на них, которые достаточны для построения гексаприм. Чтобы понять, почему теореме Паскаля, а следовательно и ей аналогичной, принадлежит основное значение, достаточно указать на то, что эти теоремы только частные выражения глубочайшей и наиважнейшей основной теоремы новой геометрии, по которой в двух проективных системах линейным совокупностям соответствуют линейные, квадратичным квадратичные, вообще совокупностям n-го порядка совокупности того же порядка. При этом пересечениям соответствуют пересечения, касаниям касания, инволюциям инволюции.
Автора заинтересовало не применение формулы для определения плоскостей симметрии, которое весьма ограничено, так как относится только к случаям, когда плоскости симметрии проходят через все оси симметрии (то есть только к зеркальным видам симметрии, когда симметричные фигуры могутъ быть воспроизведены в гоноэдрических зеркалах), а сама формула с ее численными соотношениями. 15 октября 1915 г. Тем же автором и в том же журнале (April 1916 р. 324) воспроизводится формула Миллера, приведенная в этих Записках (V 233). 4 мая 1916 г.
Цель этих препаратов—дать возможность во всех случаях, когда лабораторные кристаллы получаются из пересыщенных растворов параллельно гониометрическому произвести и оптическое исследование с возможно точною opиентировкой осей оптического эллипсоида. До последних лет я употреблял самодельные препараты, составляя их из обрезков покровных стеклышек, так нарезанных и наклеенных на предметное стеклышко, чтобы образовалось требуемое клиновое пространство для кристаллизации. Теперь я остановился на сравнительно простом и очень удобном для исследования типе, который вполне уясняется прилагаемым планом и разрезом микроскопического препарата (см. статью).
Я имею в виду ограничиться лишь более простыми выводами, употребляемыми мною на лекциях (см. статью). Из этой тройственности мы выводим необходимость ограничения для главных чисел символа комплекса, то есть, что для октаэдрической структуры нельзя принимать число, большее чем 63 1 / 2 °, для гексаэдрической можно принимать лишь числа в пределах 45° — 63'/ 2 °, а для додекаэдрической только числа больше чем 45°.
Эти немногие примеры (см. статью) дают в высшей степени важные указания на зависимость кристаллизации от расположения атомов в частицах сложного строения. В таких частицах, следовательно, необходимо отличать центральную и периферическую части (в простых частицах напр. ClNa это невозможно) и только последняя плотностью расположения атомов определяет положение возможных граней.
По сущности основного закона кристаллохимии положение каждого отдельного атома выражается в координатах (см. статью), где как числители, так и знаменатели целые числа, и притом знаменатели непременно числа, большие чем числители, если точки находятся внутри элементарного параллелоэдра или на его поверхности. Пользуясь результатами, мы по символам трех точек, легко уже непосредственно можем получить символ проходящей через них грани.
(По поводу книги Bowen The later stages of the evolution of the igneous rocks). Автор этой заметки давно различал между изверженными породами нормальные с более или менее строгими признаками химического равновесия и аномальные, не поддающиеся никаким законам равновесия и никакой строгой научной классификации, а несущими на себе яркие признаки последовательного хода явления, который, благодаря им и может быть выяснен в своей последовательности. Такие породы можно только описывать, а из описания выводить историко-геологические даты. Теперь, после опубликования важного труда Боуена невольно склоняешься к мысли о весьма слабой представленности нормальных пород и пожалуй даже их отсутствию; в природе представлены только приближения к ним, почему на них скорее приходится смотреть не как на нормальные, а как на идеальные.
В статье «Результаты первой стадии экспериментального исследования структуры кристаллов» (в примечании на стр. 361) установлен принцип определения структуры по параллелоэдру наименьшего объема, равносильный принципу установления системы параллелоэдров наивысшего порядка, допустимого для данной правильной системы точек. Критерием правильности построения параллелоэдра может служить испытание возможности иного, более специального, расположения одного из атомов представленных в химической формуле в наименьшем числе, или, еще лучше, если нет единичных атомов, помещение той специальной точки, которая занимает в параллелоэдре единичное положение.
Первое, что следует из сделанных наблюдений, это несомненная связь образования актинолита с разложением плагиоклаза. Ведь эта связь до того неразрывна, что, как упомянуто, авгит решительно нигде не прикасается к плагиоклазовым псевдоморфозам. Нам, конечно, неизвестна ближайшая первая причина разложения плагиоклаза и авгита и конечно, она сводится к остающемуся нам неизвестным содержанию составных частей проникавшего в породу минерализованного раствора, в состав которого, однако, должны были входить как СаО (от разложения плагиоклаза), так и К2О (иначе не образовалось бы мусковита), и при этом первая часть диффундировала из плагиоклаза, а вторая по направлению к нему. А так как в местах выхода диффундировавших частей из очертаний плагиоклаза мы видим превращение авгита в актинолит, и так как именно СаО есть единственная, исходящая из плагиоклаза, составная часть, содержание которой в актинолите больше, чем в авгите, то приходится заключить, что именно она и останавливалась и поглощалась авгитом, также задетым общим химическим изменением породы, более резким в плагиоклазе, чем в авгите.
По знаменитой теории Штейнера двумя данными на плоскости инволюциями пар точек на прямых определяется инволюция на любой прямой на плоскости то есть полная секунда инволюции. Определяющим фактором всех этих инволюций является линейная прима кривых, а именно коноприм (Kegelschnittbüschel по Штейнеру), имеющих общие две пары точек, из коих не только одна, но и обе могут быть мнимыми. Каждая прямая пересекает каждую кривую примы в пapе точек принадлежащей ей инволюции. Специально мы можем определить коллинеации двумя осями без всяких инволюций. Если оси, точки коих есть вещественные двойные точки всех инволюций, мы назовем вещественными, а оси изотропных инволюций назовем мнимыми, то получим, что всякая осевая коллинеация может быть определена парою осей, вещественной или мнимой (см. статью).
Если через какую-нибудь точку на квадратичном цилиндре мы проведем секущую плоскость (определяющую коноприму к), плоскость касательную и в ней какую-нибудь косую прямую d (не в плоскости кривой к и не производящую цилиндра), а затем из каждой точки этой прямой проведем в диаметральной плоскости цилиндра два луча через точки кривой к, то получим линейчатую поверхность 3-го порядка.
Системы точек и лучей на плоскости, как известно, не родственны. Но системе точек родственны секунды коноприм точек и лучей, имеющих три постоянные элемента, а из имеющих два постоянные элемента родственны только конопримы точек (тетрапримы). Теперь мы покажем, что секунды парабол лучей с двумя постоянными лучами родственна система лучей на плоскости.
Проективные отношения, выясненные в предыдущей заметке, получили бы высокую степень наглядности, если бы удалось установить простым построением такую коррелятивность между конопримами секунды и точками плоскости, чтобы экстраэлементам одной соответствовали экстраточки другой. В секунде коноприм лучей заключается прима таких, которые представлены парою точек (точнее парою линейных прим лучей). Так как в каждой линейной приме таких особых коноприм представлено три (из коих одна пара может быть мнимою), то совершенно очевидно, что приме особых коноприм лучей коррелятивна кривая 3-го порядка.
В предыдущей заметке мы разрешили задачу переноса всякого рода построений, произведенных на плоскости, в систему коноприм точек и лучей; и обратно, мы свели решения всякого рода задач в секундах коноприм точек и лучей, имеющих три постоянные элемента, к обыкновенным задачам на плоскости. Естественно является мысль распространить решение и на те простейшие случаи, когда в секундах коноприм имеются только два или всего один общий элемент.
На какой-нибудь коноприме мы можем произвольно взять две группы точек, по четыре в каждой, и по ним установить коллинеарность общего характера (см. статью). Цикл может состоять из различного числа точек, вплоть до бесконечности. Если например точка в коллинеации самогомологична, то весь цикл состоит из одной единственной точки; если имеем двойную гомологичность точек А и А', то весь цикл сводится к двум точкам и т. д. В общем случае цикл обнимает значительное число точек или даже их бесконечное число, и может случиться, что все точки конопримы входят в состав одного цикла. Если коллинеация, как-нибудь установленная по данным точкам, делает коносекунду самогомологичною, то задача построения точек последней, сводится к простой задаче коллинеарных построений.
В заметке „Полярные отношения мнимых трехугольников и четырехгранников" мы показали, что эти отношения тождественны с теми, которые определяются известным мнимым эллипсом или эллипсоидом, причем ни тот, ни другой не проходит через данные точки. Но из оснований, приведенных в этой заметке, следует, что могут существовать определенные полярные отношения вещественных трехугольников и четырехгранников. Эта работа составляет естественное продолжение предыдущей заметки, но относится к трехугольникам и четырехгранникам принятым за вещественные.
Выведя ряд главных совокупностей, как позиционных, так и непозиционных, мы можем теперь более отчетливо определить само понятие таковых. Главною совокупностью мы называем таковую, которая вполне и однозначно выводится по данному числу элементов, если всем этим элементам принадлежит в построении одинаковая роль. Если означим данные элементы буквами и выведем построение совокупности, вводя сначала элементы, отмеченные одними, а потом другими буквами, то если выводимая совокупность есть главная, то мы можем буквы по отношению к элементам переместить иначе, а построение остается справедливым, если в его порядке хода мы оставим, и прежние буквы.
Хотя предыдущая статья и представляет собою нечто законченное, доведшее главные совокупности до тех, которые определяются шестью элементами, но сразу бросается в глаза, что высшие из них— гексасекунды не представляют собою совокупностей наиболее общего характера—коносекунд, а только специальные их разности, могущие быть воспроизведенными прямыми то есть только линейчатые коносекунды. Только таковые вполне и однозначно определяются не более чем шестью точками и плоскостями. Так как именно коносекунды общего характера устанавливают полярную коррелятивность между точками и плоскостями в пространстве, то доказательство теоремы, о которой идет речь, сводится к тому, что такая коррелятивность может быть установлена ∞ 9 способами.
Известно большое число особых точек, положение которых строго выводится для каждого данного трехугольника. Эти точки открываются при изучении тех или других свойств трехугольников, число коих весьма значительно. Несмотря на всю простоту такого образа, как обыкновенный трехугольник, изучение его все-таки еще не может считаться исчерпанным и с течением времени, хотя и редко, открываются новые и новые свойства. Представляется задача найти ту точку, которая служит центром линейной примы лучей—поляр точек описанного круга. Для решения этой задачи достаточно найти, поляры двух каких-нибудь точек круга по отношению к трехугольнику; точка их пересечения и есть искомая. Она то и составляет ту новую особую точку трехугольника, о которой говорится в заглавии.
Как известно, эти поверхности IV порядка могут быть воспроизведены двумя проективными квадратичными примами плоскостей. Teopия этих поверхностей излагается в подробных руководствах. В общем случае на поверхности имеется гексаприма точек, в которых пересекается по два луча поверхности. Плоскости, проходящие чрез такую пару лучей, могут считаться касательными плоскостями, а так как плоское сечение поверхности в общем случае есть кривая IV порядка, то понятно, что если плоскость проходит чрез один из лучей поверхности, то эта кривая распадается на этот луч и кривую III порядка, а если плоскость проходит чрез пару лучей, то та же кривая распадается уже на эту пару и еще коноприму; отсюда видим, что касательные плоскости, проходящие чрез пары лучей поверхности, пересекают эту поверхность еще в коноприме.
В статье о линейных совокупностях лучей я показал, что система лучей не есть система самостоятельная, но что для нее нужно принять параметр в виде экстралуча, необходимо входящего в состав линейных совокупностей. Тогда линейная прима определяется вполне и однозначно двумя, линейная секунда тремя и линейная терция четырьмя произвольными лучами. Если даны три произвольные луча, то совокупно с постоянным четвертым экстралучом мы получаем необходимые и достаточные данные для определения лучей полной линейной секунды. Так как четыре произвольные луча в общем случае пересекаются парою секущих, вещественною или мнимою, то ясно, что линейную секунду мы можем определить не иначе, как совокупность лучей, пересекающих данную пару прямых а и b . Для простоты примем ее вещественною. Означим экстралуч, определяемый экстраточками этих двух общих секущих, чрез е.
По определению этой поверхности она воспроизводится двумя гомологичными квадратичными примами плоскостей. В общем случае на этой поверхности имеется гексаприма двойных точек, в каждой из которых пересекаются два луча поверхности. Коррелятивное преобразование дает такую же поверхность, так как каждой точке с двумя пересекающимися лучами, коими определяется касательная плоскость, коррелятивная касательная плоскость с двумя лучами в ней, пересекающимися в точке касания. Следовательно, такую поверхность мы можем воспроизвести и коррелятивным путем то есть определить ее двумя конопримами с установленной проективностью точек.
Терции лучей, представляющие так называемые нулевые системы и вполне и однозначно определяемые пятью произвольными лучами, обыкновенно называются линейными в виду того, что в каждой плоскости и из каждой точки пространства имеется прима лучей, пересекающихся в одной точке и заключающаяся в одной плоскости; такая прима лучей в плоскости или исходящих из одного центра называется линейною. Но для системы лучей, определяемой, как параметром особым экстралучом, такие примы уже не есть линейные, так как последние необходимо должны заключать в себе этот экстралуч. Поэтому и сами нулевые системы не представляют в этой системе линейных терций, а есть терции квадратичные.
Ряд последовательных усовершенствований в точном изображении химического состава горных пород показал, как найти фигуративную точку химического состава по данным четырем отношениям окислов. Однако пространственное положение этой точки определяется не одною, а двумя проекциями на взаимноперпендикулярных плоскостях по методу начертательной геометрии. Хотя во всех других отношениях была достигнута высшая достижимая простота, но все-таки наглядность в изображении несколько страдала именно вследствие изображения в двух проекциях. В этой статье я имею в виду систематически изложить ход всех операций, необходимых для графических изображений так, чтобы они стали ясны даже для лиц, не имеющих понятия ни о тетраэдрической схеме, ни о системе векторальных кругов.
В третьем выпуске „Универсального Метода Федорова", в главе VII—„исследование дисперсии"— В. В. Никитиным описан идокраз, обладающий резкой дисперсией двупреломления. Кроме того он обладает еще и тем свойством, что в пределах одного и того же зерна, в различных его участках, величина двупреломления различна. Благодаря этому интерференционная окраска не однообразна на всем протяжении зерен идокраза. В сечениях близких к параллельности оптической оси—четверной оси симметрии кристалла—цвета располагаются полосами, параллельными друг другу: центральная часть зерна имеет белую интерференционную окраску, за ней следует полоса с мастично-желтой окраской, затем—с пурпуровокрасной и, наконец, края зерна окрашены в пурпурово-фиолетовый цвет.
Ю. В. Вульф любезно обратил мое внимание на весьма маленький, но образцовый учебник Миллера Tract on crystallography, вышедший в Кембридже в 1863 году. Это книжечка всего в 86 страниц, но тут не только перечислены и изображены главнейшие формы кристаллографа, но, что для нее особенно характерно, приведены и выведены главнейшие формулы для вычисления и притом по той оригинальной для автора системе, в которой преобладают двойные (ангармонические) отношения. Формулы Миллера впервые ввели в практику вычислительной кристаллографии начала новой геометрии, хотя еще их вывод целиком основывается на формулах плоской и сферической тригонометрии.
П. П. ф. Веймар любезно прислал мне скляночку с крупными кристаллами при письме: «На дне склянки, куда сливались реактивы при мытье посуды образовались кристаллы зеленого и желтого цвета; эти кристаллы я Вам посылаю; быть может они интересны для Вашего кристалло-химического анализа». Кристаллы разных цветов оказались и разной величины и разного облика. Желтые кристаллы ясно пластинчаты; толщина пластинок чуть не в полсантиметра, а наибольший их размер превышает два сантиметра. Зеленоватые кристаллы представлены в более изометрическом виде и по линейным размерам по крайней мере в четыре раза меньше.
The general law discussed here is that a crystal falling out of solution tends to take on the smallest surface area. This law, which has a simple and well-known theoretical basis, is usually demonstrated by examples of crystallization, or rather recrystallization, which requires a long time, even months, or at least days. I have come across a preparation in which this demonstration can last for several seconds. This drug is sodium nitrate, microscopic crystals of which dissolve from breathing in a few seconds and crystallize at approximately the same time due to evaporation. Thanks to this speed, of course, the mentioned law сan be also quickly demonstrated.
Involutory-collinear transformations are among the most elementary operations of new geometry. But this always means the transformation of real geometric images. The problem of transforming imaginary images, for example, imaginary circles, as it seems, has not been posed and seems incomprehensible. When studying a system of pairs of rays, this problem was presented in all its reality in the following form. If two pairs of rays are given, we take them to be two pairs of tangent parabolas, which is easy to construct, and thus we find a linear primе of pairs of rays whose centers form a straight line, and the pairs of rays themselves are pairs of tangents to the parabola.
One of the very first principles of the new geometry is the theorem according to which projectivity on primes (linear and square) is established by the correspondence of three elements. Therefore, if four arbitrary lines are given on a plane, then each of them, in intersection with three others, gives three points, and this is enough to establish the projectivity of points on all these lines, because on each of them we have three corresponding points. If spherical aggregates are given partly by real, partly by imaginary conoprimes, then from them it is necessary to construct two linear aggregates of the same level, of which for one it is necessary to change the value of the category of conoprimes: take the real for the imaginary and vice versa.
The most important categories of hexaprimes, or what is commonly called spatial curves of the 3rd order, were derived by Seydewitz and are given in the famous manual by Reye under the names 1) spatial hyperbola, 2) spatial ellipse, 3) parabolic hyperbola and 4) spatial parabola. This note was the result of a task: is it possible to build a hexaprima that has symmetry? The term hexaprima means such a prima of points that is completely and unambiguously determined by six points, and a 3rd order spatial curve is just such a curve. We get three constructions leading to hexaprimes of three types of symmetry (see article).
“In my previous works, I examined a number of geometric systems, the elements of which consist of pairs of points. The simplest and most important of them is the system of parallel vectors, then the system of harmonic segments and vectors, and finally the system of midpoints of harmonic pairs. But in all these systems considered, some limitation is introduced either in the form of vectorality or in the form of a special parameter of the system. Here I mean to consider a system of such elements, given without any restrictions, that is, I imagine that an element of a system on a plane can be an arbitrary pair of its points, which at the same time will form a segment.
This theorem very simply solves the problem, which can be formulated as follows as a problem of elementary geometry (see article). Despite the simplicity of solving it as a problem in a system of circles, it is hardly solvable on the basis of theorems of elementary geometry. It is clear that this theorem can be directly transferred to the system of spheres by replacing the words “circle” with the words “sphere”. To prove it, it is enough to take the central Q R as the axis of rotation. Thus, in the most general form, the problem of finding the centers of spherotions of sphers, which was previously solved through formulas, is resolved.
In previous works, I developed a vector system in detail. But since vectors essentially represent pairs of points, albeit unequal (initial and terminal) and since correlative rays can be established for each point, it is clear that systems of vectors can be correlative to systems of pairs of rays that can hardly be called anything other than vectorial. But if it is somewhat difficult to talk about vectorial pairs of rays on a plane, then, for the sake of complete clarity, I prefer to establish such a sequence of systems that are part of all the linear elements of this system.
We have come to the conclusion that the system of rays can be introduced into the general series of geometric systems, and we can bring it into a general correlative connection. However, this system no longer belongs to the number of independent ones, but to the number of systems limited by a certain parameter, for which we must recognize a certain extra-ray, constant for all linear primes just as we have a similar system of points with a parameter point included in all linear primes of this system.
Any general collection of circles will not differ from the collection of circles of the previous system, but will constitute only half of the collection of this system, and the linear primes and seconds of ordinary circles will remain the same for this system; but the linear collections of vector circles of the previous system will no longer be such for this system , because the tangent linear primes of the previous one are no longer the linear primes of this system. It is easy to prove that in this system there are no collections of vector circles at all; they cannot even be specified. In fact, if I think about, for example, a right vectorial circle, then diametrically opposite to it is already a left vectorial circle; in essence, two vectorial circles are obtained, which completely and unambiguously determine their linear primacy on the sphere; it is clear that in her presence it is impossible to set a third, arbitrary circle; in general, it would no longer be part of a certain linear prima.
The article “A simple and more accurate representation of points in 4-dimensional space on a plane using vectors” not only examines in detail the system of vectors on the plane, but also indicates the basis for constructing a linear prima of vectors in space based on two data, namely, that this linear prima consists of segments of the generating hyperbolic paraboloid, enclosed between two guides, of which one is the line of initial points, and the other is the line of end points, and the construction itself can be done by decomposing two given vectors into a component along three coordinate axes and constructing linear parallel vectors from the components; three vectors with a common starting point, but parallel to the coordinate axes, no matter what directions we choose for the latter, are the constituent vectors of a linear prima; the end point of the latter is on the line of end points.
The use of X-rays enabled V.L. Bragg (and his father) to get the means that led to conclusions extremely important for the theory of crystal structure. These conclusions are partly unexpected, at least in the sense that it was expected to see the centers of chemical particles at the points of regular systems, while the experiments of the named scientist led to the conclusion that these are the centers of atoms. Thanks to this, in substances of the simplest chemical composition, special regular systems of points are obtained, and the centers of symmetry are occupied by individual atoms, as if the atoms themselves also have high symmetry.
This note is motivated primarily by the desire to present Bragg's final conclusion in a more visual form; and then, in view of the complete originality of this conclusion and the rather sharp discrepancy with our previous ideas about the structure of particles, there was a desire to decide whether it was possible to reconcile it with them. Thinking about the arrangement of atoms, we can easily understand that it is of two kinds. Some atoms occupy the positions of the centers of rhombic dodecahedrons, others occupy the positions of such four trigonal vertices of the dodecahedron that together they belong to the tetrahedron. It is this arrangement that determines the hexakis-tetrahedral type of symmetry, and although the location of the centers of some particles corresponds to the dodecahedral structure, the situation is changed by the arrangement of other atoms.
Strictly speaking, projections can be constructed as many as geometric systems of the second stage, that is, an unlimited number, and if I now want to mention such as new crystallographic ones, it is solely because they represent a kind of convenience for solving certain crystallographic problems not delivered by other projections. Here I mean those projections that are obtained from the linear and gnomonic ones if they are subjected to transformation by inverse radii, which is why they can be called grammatic and gnomocyclic, respectively.
In previous works, methods for such a determination were given using the tables of Sokolov and Artem'ev, with the exception of the cases listed in the title, unless the pole corresponding to the double axis of symmetry is also the pole of the face (1000). In all these cases, it is assumed that certain displacements are made to determine density of the main belt faces. In this note I will show that for these cases you can do without a displacement (see article).
Though from the previous note, one can see how important, when initially studying the cristallography, is the derivation of special simple formulas for determining the density of isotropic complexes, cubic and hypohexagonal. The beginners are best and most easily acquainted with the technique of determining densities from tables precisely on examples of isotropic complexes, since simple formulas give perfect control to the determinations made and immediately familiarize them practically with the degree of accuracy or rather inaccuracy of graphical operations.
The known properties of the gnomonic projections of trigonaloid crystals prompted me about the presence of the relationships mentioned in the title that seemed paradoxical to me. For the case under consideration, the theory of poles and polars unfolds in its usual form: two points are the poles of two polars and, in turn, determine the straight line-polar of the point of intersection of these polars. Each vertex of a trigon has a polar opposite side, etc., and in no case there is a point through which its polar would pass, as it occurs in the case for imaginary projective conjectures (see article).
Pascal’s theorem underlies the doctrine of conoprimes, expressing their fundamental property that is completely and unambiguously determined by five elements. It can be expressed in a modern generalized form (see article). This expression clearly demonstrates the deep organic connection of every sixth element with the other five that determine conoprima. A simpler analogue of this theorem can be the well-known theorems expressing the fundamental properties of spheroprimes and spheroseconds.
The content of this note is a direct consequence of the previous one. It contains a theorem that makes it possible to construct a hexasecond using seven arbitrary points. Since the hexasecond is a positional image and is correlatively transferred to the entire geometry of the system, then the construction of a hexasecond of planes based on seven data is implied by itself. But if only six planes are given, then the extraplane is always, like the seventh, at our disposal and is not included in the count as the only one of its kind.
If we take into consideration only a plane on one side, and a homological hyperboloid on the other, then even in this case we can recognize the relatedness of these linear seconds of points, because the infinitely distant points of the first system, and therefore their complete linear example, are homologous to the points of the straight line of intersection of the hyperboloid with the plane of involution, and consequently this entire straight line as a linear element of extraelements. From here, in particular, it follows that if three arbitrary points are given in the second system, then the spheroprim defined by them can easily be obtained in this way: we project these three points through the center Z onto the plane, construct a circle based on them, and transfer the points of the latter to a parabolic hyperboloid by inverse projection . The center of this spheroprim is also designed. It is clear that this center on the hyperboloid in relation to the spheroprim is the pole of the mentioned straight line on the hyperboloid.
When starting a goniometric study of a crystal, the researcher cannot yet foresee how the crystal will have to be expediently oriented on the goniometer in order to, at the end of the work, in the simplest way, from the resulting diagram, derive the symbol of the complex—the first goal of any goniometric study. From here it is clear that in general, when the question of the correct installation of the crystallographic complex is resolved, it is finally resolved only after completing a series of measurements, deducing the most significant faces and calculating the planes—it is necessary to transform the projection plane, taking the main belt as the projection circle.
I take this opportunity to note for the thousandth time the advantages of the method of new geometry, the theorems of which know no exceptions, but always have perfect generality. This author has already given the proof of this theorem, which I communicated to him, using the method of new geometry. It seems to me that A.K.’s theorem is much more expedient and simpler. Boldyrev should be formulated as follows (see article). Theorem of A.K. Boldyrev in a generalized form, using the method of new geometry, reveals one of the interesting properties of spheroprims of vectorial circles.
With regard to the theory of confocal sets, the conclusion drawn shows that the set of surfaces derived from the imaginary hyperbola taken as the focal curve does not represent anything new, and was included in those that were derived on the basis of the real hyperbola. If we take into account that in the general case we have two focal curves connected by the main axis on two mutually perpendicular planes of symmetry, one of which is an ellipse and the other a hyperbola, that on the third plane of symmetry the focal curve can be neither an ellipse nor hyperbola, and, as it now turns out, an imaginary hyperbola, then it remains possible to assume only an imaginary ellipse, which is where the derivation of focal curves ends. In conclusion, we note that it is possible to derive involutions on the plane at infinity; since for it three normally conjugate rays are projected from any point, then the corresponding projectivity curve is an imaginary circle, and this is the case for all confocal populations in space.
In this article I present, however possible, a complete list of crystals of the cubic system obtained to date. Here we are really dealing with a number of substances that are exceptional in their properties, like exceptional forms of the cubic system among all the others. When the list is compiled, this exclusivity is in the chemical composition of the substances What is striking is the possibility of dividing them into those few headings that form the basis of my presentation. Other features of the chemical composition of substances in this series will be discussed at the end of the article. I needed to compile this list simply because from those described, I could single out those crystals that cannot be determined by the method of crystal chemical analysis.
В статье поднимается вопрос построения ребер по символам в кристаллах для комплексов гипогексонального типа, который пока еще никем не поднимался. Разрешается же он, конечно, чрезвычайно просто и притом вполне аналогично с разрешением его для кубическаго типа. Мною было показано, что в комплексах гипогексагональнаго типа символы ребер таковы, что в частном случае гипогексагонально-изотропного комплекса индексы ребер и перпендикулярных граней одни и те же, как это и требуется учением о сингонии (потому что в этом случае эллипсоид сингонии есть шар).
In a gramstereographic projection, every plane is projected by an arc of a great circle, that is, an arc passing through two diametrically opposite points of the projection circle. This circle represents one circular section of a cone having a center at the point of convergence of the rays; another circular section of the same cone is the diametrical circle of a sphere at projected plane. Apparently, not a single crystallographer has yet noted that these projecting cones are not cones of a general nature, but are special cones, called cones of Pappus, who was the first to note their simple construction. Both special axes of the projecting cone are perpendiculars to both circular sections, that is, perpendiculars to both the given plane and the projection plane (see article).
In these Notes (III 287) I devoted a note to this subject, in which I limited myself to only a complete derivation of the geometric images related here. The same conclusion, of course, could be produced in other ways, which should lead to identical results. All conoprims, for example, can be derived by a collinear transformation from circles, but also by notching two projective prime rays, and from elementary manuals it can be seen that the last example, if not more correct, is at least more visual, and in this sense simpler (see article). From the note mentioned at the beginning, it is clear that the existence of special circles and spheres brings great disruption to the ideas about the circle that have been established, not even for centuries, but for millennia. As it is proven in it, the idea of a center and equal radii must be completely eliminated from this concept."
Difficulties in crystallographic practice include difficulties in compiling tables of crystal chemical analysis and the actual research techniques. The author formulates the course of operations necessary for a complete geometric study of crystals of one substance, assuming that the goniometer is calibrated and satisfies the conditions stated above (see article). Before each individual series of measurements, one should start by setting the zero. The first measurement, leading to the compilation of a diagram in stereographic projection, serves for a closer acquaintance with the crystal and the selection of the main faces. If the substance has already been described, then often this first measurement makes it possible to derive the correct installation and the associated symbol of the complex, and therefore to determine the substance from the tables.
If we are given a type of symmetry and, in accordance with it, on one given sphere we place equal spheres in layers at the distance of their centers from the center of a given sphere and, moreover, in such a way that these spheres fit into the recesses between the previous ones and form a regular set, then the number of spheres of the layer will be quite definite, namely, it will be equal to the value of symmetry in the general case, when the direction of the radius vector of each such sphere (starting from the center of the initial sphere) will be common (that is, neither coincides with the axes of symmetry, nor is in the planes of symmetry), and will a certain divisor of this number in particular cases. I will consider three sets of spheres of hexakisoctahedral type of symmetry, corresponding to the three possible systems of parallelohedrons in this case: triparallelohedrons, hexaparallelohedrons and heptaparallelohedrons, as well as a set of dihexonic-bipyramidal type of symmetry (and a system of tetraparallelohedrons).
Imaginary circles play a very important role in solving geometric and crystallographic problems. Imaginary spheres also play a huge role. But when solving some problems related to some simple sets of imaginary circles and spheres, as problems of an elementary geometric nature, difficulties arise that are easily eliminated precisely thanks to the simple properties of these sets. The article discusses possible ways to solve these problems.
If, taking the center of the cone as the center of the sphere, we intersect it with the surface of this sphere, then the cone will be replaced by a spherical coneprime, which is why the task at hand is reduced to recognizing the ranks of coneprime on the sphere. A similar problem of determining the ranks of flat conoprimes is resolved by determining the type of involution of the points of the conoprime on an extra straight line (infinitely distant) or the involution of rays at its center. The article introduces new ranks of conoprimes and a method for recognizing them.
The numerous and useful results that came from drawing up a diagram of spherical conoprimes prompted me to take up, as a simpler case, drawing up a diagram of flat conoprimes. Of course, in both cases the difference is enormous. There we are dealing with the second of conoprime; here only with prima, since the totality of all similar conoprims has to be considered as one single one. There, each conoprim is characterized by the angular magnitude of two axes, which are always real; here only the main (major) axis is always real, while the minor axis in hyperbolas is the imaginary axis. The diagram is based on combining all similar conoprims into one. But in the composition of hyperbolas there is a striking exception in relation to similarity, namely the extreme difference of hyperbolas with equal angles between asymptotes, that is, the pair of asymptotes itself, like a hyperbola, cannot be called similar to all the others. For this reason, the diagram does not include special hyperbolas consisting of a pair of rays.
The author compiled the accompanying diagram, using, as for the diagram of spherical conoprims, a stereographic grid, excluding small circles from the latter. The diagram of conoseconds, as well as the diagram of flat conoprimes, is built on the principle of similarities, that is, all similar conoseconds are taken as one. The main purpose of the diagram is to determine, in relation to the three main axes of the conosecond, those three conoprimes that are formed in the conosecond in the three planes of its symmetry.
The formulas of spherical tetragonometry are considered, which are also applicable to plane tetragonometry. For practical purposes of crystal chemical analysis, the graphical techniques used are quite sufficient, despite the inaccuracy associated with them. But over time, as the material expands, the need for replacement of roughly obtained numbers with more accurate, which in many cases will reduce the increasingly complex labor of finding in the tables the substance defined as the symbol of the complex. A closer acquaintance with the upcoming problem shows that here it is not just necessary to solve spherical trigons at three given angles, which is precisely what is exhausted by spherical trigonometry, but that here it is possible to calculate spherical elements, in an indefinite number obtained by constructing from given four points, and find for each such element a corresponding formula expressing it even with an arbitrary change in the position of the four main points.
Experience has shown that at present, the approximate numbers that are obtained using relatively crude graphic techniques are quite sufficient for the individual characteristics of each substance, that is, for crystal chemical analysis. But as new crystallographically described substances accumulate, and such accumulation proceeds at an accelerated rate, the time must someday come when greater accuracy will be required in expressing the results of measurements.
It was to this task that Shteiner dedicated his famous work on the linear primas of conoprimes (Kegelschnittbüschel) and it was in it that he presented it with such exhaustive completeness that there would be absolutely nothing to add to it, if only he had not limited his task in advance to material conoprims; Between those data there may be imaginary conoprimes, even if in essence they are only ellipses, since imaginary hyperbolas are equivalent to real hyperbolas with the same asymptotes, the so-called conjugates.
In the very foundations of the new geometry lies the concept of involution and two cases are distinguished: involution with a pair of real (hyperbolic) and a pair of imaginary (elliptic) double elements. In particular, how for points on a line we can move from one type of involution to another if one from the systems of points that make up the involution, we turn it over so that the points that were conjugate to themselves (double) become conjugate to each other, so in the involution on the plane (polar system), the conoprime that determines the involution becomes imaginary (see article). The difference between the real and imaginary conoprime is clarified, as well as the real and imaginary conosecond. The example shows polar relations for all imaginary conoprimes and conoseconds.
By a strange accident, not a single geometer, as far as I know, was given a system of spheroprimes of rays, while the system of spheroprimes of points was one of the first established geometric systems, except for the system of spheroprimes of both points and rays, which was so ahead of its time, derived by Mr. Shteiner. At the basis of the system construction is a linear prime; and now we will set ourselves the task of constructing a linear prime of spheroprimes of rays.
In the article „Parolleloëder in kanonischer Form und deren eindentige Beziehung zu Raumgittern” I developed the concept of parallelohedra in canonical form or simply canonical paradelohedrons, taking as the main thing the duality of the derivation of these parallelohedra from spatial lattices so that this conclusion was unambiguous. However, in this article I examined only one side of the question, related to these angular relationships of crystalline complexes, which, in accordance with the crystallographic law of limits, bring all crystallographic complexes in general closer to ideal types. They characterize the distribution of angles, which determines the belonging of these types to certain types of syngony, and in them the first role is played by right angles (see article).
In the article “Paralleloëder in kanonischer Form und deren eindeutige Beziebung zu Raumgittern” I showed that by performing a monoclinic shift, you can always obtain an identical spatial lattice, and at the same time eventually you can reduce anorthogonality, that is why the expression for the probability of correct installation will increase, though the symbols of the shapes become more complex. But I did not dwell in detail on the criterion for exactly what shift the lattice remains identical. This is the explanation I want to make in this note.
The more processed material is accumulated according to the calculation of correct installation, the more pronounced is the need to limit this calculation to the minimum number of important faces. Laboratory crystals are most often distinguished by the minimum number of developed forms. This was confirmed in the example of barite. Despite their appearance, which is almost no different from the usual appearance of many natural barite crystals, their combination is minimal and through this the first most important facets are especially emphasized.
This note contains neither any significant innovations nor any systematic solution to graphical issues. But with the widespread development of graphic solutions that crystallography has received recently, and especially with the introduction of crystal chemical analysis, the most insignificant simplification or reduction in techniques acquires very significant practical significance. Finally, some rules that lead to a reduction in graphic operations in a particular specialty, their applications only in certain, although numerous, cases, are inconvenient to place in elementary courses, where only the rules of the most general significance should be considered, and, moreover, presented systematically, so that students receive a valid ability to solve problems of all kinds, even if not always in the simplest way.
In view of the fact that tables for this analysis have now been compiled, it was possible to begin the application of this scientific discipline. A total of 5 tables have been compiled, namely, all tetragonaloid crystals are divided into 3 tables according to structures (hexahedral, dodecahedral and octahedral), and in addition one table for hypohexagonal and trigonaloid crystals (for the latter, structures are noted only for ideal crystals due to their significant accumulation). In all tables, ideal crystals are highlighted in a special column, and it is for them we have the most dense arrangement of points, which is why, also bearing in mind the inevitable inaccuracies in the graphically obtained constants, for such crystals in particular we will have to compare the largest series of crystals.
Enigmatic faces can practically be viewed as irrational, not allowed by the basic laws of crystallography. The assumption of their irrationality is also confirmed by their overgrowth upon contact with an important face of the complex, moistened with a saturated solution of the substance, and part of the last face is subject to dissolution. In addition to this characteristic, enigmatic facets are characterized by their isolation and uniqueness.
This amendment is made by me to the note “An interesting crystal of apatite, a satellite of neptunite from California.” (Notes of G.I. II 253) based on the written instructions of Dr. Slavik from Prague, who noticed my error based on the data from my own description, mainly pleochroism. Having tested the hardness, which turned out to be slightly higher than thsat of orthoclase, I can now declare the error that occurred with full conviction.
In the last article “Chemical relations of rocks and their graphic representation” I focused on the tetrahedral representation method, which I proposed even earlier, as the most perfect and simple method. And at the present moment, with a view to further simplifying the same method, I focus on it as the most perfect. Thus, the purpose of this note is not to consider theoretical issues using this method, which I did in the mentioned article, but there is not even a thought to introduce any change in its applications. Everything that is developed in this article remains equally correct for me at the present moment. Now I only mean to show that the method proposed then can be applied in a wide variety of forms and to choose from them exactly the one that is associated with the simplest operations.
If a trigon ABC is given and we define the height point D in it (that is, the common point of intersection of the perpendiculars from its vertices to opposite sides), then ABCD can be taken as complete quadrangles with pairs of opposite sides AB with CD, BC with DA and CA with BD . Drawing a circle through the bases (a1, b1, c1) of the perpendiculars on the sides of the triangle, we obtain a Feuerbach circle, which, in addition to these three points, will pass through six more midpoints of the just listed sides of the complete quadrilateral, that is, points a1, b1, c1, a'1, b'1, c'1.
In my article on the system of spheres, I outlined their linear and spherical aggregates, the collinear and reciprocal transformation of these aggregates, but did not touch at all the special circles found in each of their linear patterns. It was only implied that among all the circles of such a prima there is a circle of infinitely large radius , and such a circle is a straight line constituting the radical axis of the prima. As if in contrast to this, in the scientific literature, starting with Shteiner, it is understood that in the linear example of circles the special one is not a straight line, but a pair of straight lines, one of which is the radical axis, and the other is the infinitely distant straight line; but I am not aware that the question of special circles has been specifically discussed anywhere. With this note I intend to fill this gap.
Let us consider systems of curves of the 2nd order (conoprim). In the system of synonyms of points, circles can be taken as extraelements, because these elements constitute a special system in themselves, and at the same time, any curve with a circle defines a linear prima. In general, in a linear prime there is no such extra element, but only in a linear second. However, you can make a linear second from the linear prima of ordinary (not vector) circles and some other conoprim. Such a linear second, however, will already be special, and therefore should be considered as a special system, and such a system will be related to the system of points on the plane, and the circles of the first must be projective in a special way to the infinitely distant points of the latter. Also, if we compose a linear third from some linear second of circles and some other conoprime, then such a system will be related to the system of points in space. But all these will be special, special systems of connotation points.
In the "Annaks of Geology and Mineralogy of Russia", in the article on “Crystallization in a solid medium,” I already described the experience of converting polyhydrate magnesium sulfate hydrate, directly formed during the evaporation of the solution, into heptahydrate. At the same time, I noted that the rapidly growing needles and fibers of heptahydrate hydrate with apparently they spread at the same speed both in a free solution and penetrate the crystals of a polyhydrate hydrate (namely MgS0₄ 12 aq). It must be assumed that such an extreme slowdown in the progress of the phenomenon in extra-thin layers occurs under the influence of partial capillary forces of extension between the walls of the wedge space and its contents.
It is clear that the complete set, that is, the fifth of conoprims, has the highest possible symmetry, that is, circular symmetry. The symmetry of quarts is completely determined by the symmetry of one conoprim, because from it it is completely and unambiguously derived. Therefore, in the general case, such a set has a double axis of symmetry and two perpendicular planes of symmetry (rhombic type of symmetry on a plane). In the special case of a parabola, only the plane of symmetry remains (the hemirhombic type of symmetry). The circle has absolutely exceptional symmetry, and therefore there are linear quarts that have circular symmetry. From here we conclude that if, to define a linear quart, we take an arbitrary conoprime and a fivefold axis of symmetry, from which five equal ones are derived, then we obtain a quart with circular symmetry. All the curves contained in it in all positions are arranged in continuous circles of equal elements.
If it turned out that by choosing one linear prima in one diametrical second and then arbitrarily another linear prima in an arbitrary other second and thus constructing an infinite number of hyperboloids, we would obtain that the totality of such hyperboloids lies in one third located in one third , located in one linear quart, then we would be dealing with an image representing a generalization of the concept of a hyperboloid; we could call such a hyperboloid a hyperboloid of the 4th stage system. The special third, which has the symmetry of a circle, just indicated in the article (“Symmetry of linear sets of conoprims”) is such a generalized hyperboloid in the conoprim system. Since in this system, which treats a completely different topic, it would be inappropriate to dwell on the consideration of this issue in all details, this note has been dedicated specifically for this purpose.
As is known, the French mathematician Dupin used the name cyclides to refer to curious surfaces that can be defined as being encircled by the set of all balls tangent to three data. These surfaces are extremely diverse and stand out for their many simple properties. studied both by the author himself and by some other mathematicians. They have two special axes, and if you rotate a plane around these axes, it will cut the surface in a continuous series of circles, which is why this surface can also be imagined as the trace of a circle moving according to a well-known law, at all points perpendicular to all circles of another similar system . All properties of cyclides are set out in my manual “New Geometry as the Basis of Drawing” (101). But here, in addition, a special cyclide with extremely interesting properties was derived.
There is absolutely no indication that the ore-bearing conditions, which in any case are evident here, will cease with the places of previous mining operations. True, there is no reason to assert that we will certainly encounter rich ore deposits here; but it must be said that everywhere and even in other places, no matter how favorable the observed conditions may be, it is risky to make positive statements, but in the conditions drawn by a compiled geological map, it is more likely to assume favorable rather than unfavorable results. The Nikolo-Podgornyi mine occupies a very special position. From a geological point of view, it is one of the most interesting points in the area, both for the distinctness in the development of the rocks, their complete exclusivity among others, and for the unexpectedness and novelty of the rocks themselves and the geological conditions in which they were formed.
Taking into account the immeasurably greater simplicity of the method of new geometry as a method of mental construction (without the help of any auxiliary complex adjustments), one does not need to be a prophet to foresee that modern geometric analysis, in pursuit of its task, will supplant algebraic analysis, and the role of the latter will be reduced to such symbolic expressing the conclusions of geometric analysis (which is necessary to replace the essentially inaccurate implementation of geometric constructions in practical applications with precise calculations and calculations), which makes it possible to express the results in exact numbers.
We can characterize collineation with an imaginary involution by two such self-collinear rays, of which one is infinitely distant in the horizontal plane, and the other is vertical. Although these two rays are truly self-collinear, and in no way are the axes of collineation with a real involution, but as two special ray, characterizing the symmetries of the system, we could conditionally call them the axes of imaginary collineation (conventional abbreviation of collineation with imaginary involution). We see that these systems have a center, three double axes of symmetry passing through it, and three planes of symmetry passing in pairs through the two axes of symmetry.
This includes, on the one hand, the substance obtained by Anschutz u. Beckerhoff as Benzoylderivat des Amyiphenols and Benzoylderivat aus Tertiaramylphenol, the crystals of which were described by Hartmann, and on the other hand, the substance Benzoyl, obtained by the same chemists, p. tertiara-mylbenol, the crystals of which were described by Schwanke.
The question posed is so elementary that, it would seem, its solution should lie in the most elementary textbooks. However, this did not happen, and in the most comprehensive manual available - Reye, Geometrie der Lage in Chapter 3 of Volume II, which specifically treats the perspective position of linear seconds, only the conditions under which two linear primes are considered (as usual, only two systems are considered - a system of points and a system of planes) are in a perspective position. Therefore, I think it useful to consider this issue in general terms.
If it is impossible to unambiguously determine an infinite set of rays from arbitrarily given two of them, then this can be achieved from arbitrarily given three of them. It is well known from elementary manuals that with three arbitrary given lines, and, moreover, non-intersecting lines, one can completely and unambiguously determine a certain unisexual hyperboloid. Since this curved surface of the 2nd order consists not of one, but of two systems of non-intersecting straight lines, it is clear that from the three straight lines only one of them is directly determined, which includes three data, and then it is logically inevitable to also accept the other a collection that occupies a position in space identical to the first system, that is, the surface of a unisexual hyperboloid.
The plane passing through the polar a and the point, has as its zero point the one at which the polar intersects with the zero plane of the point A. The straight line connecting this point B with point A, like a polar, has a point on polar a as its pole, and both of these points make up conjugate pair on this polar. Each plane, simultaneously tangent to two conoseconds of such a prima, has as its polar a straight line connecting the two points of tangency. If the plane is simultaneously tangent to more than two conoseconds, then it is tangent to all conoseconds of the linear prima, which in this case have one common point of tangency with it and with each other. The zero system is polar with respect to linear primas conoseconds, just as the ordinary polar system follows from a single conosecond.
Recently I noted the likelihood of the identity of two substances obtained by Anschutz and described one as Benzoylderivat des Amylphenols and the other as Benzoylparatertiaramylphenol. After the publication of this note, I received a kind letter from Prof. Grote, who inquired about this in writing from Prof. Anschutz and received an answer about the actual identity these two substances, despite the enormous differences in the crystallographic constants assigned to their crystals by Messrs. Hartmann and Schwantke. Now I have found in my old notes an indication of the same ratio of two substances, which are given very different chemical formulas, but which in crystallographic terms turn out to be very close.
The geometric constants, even for a triclinic crystal, are five angles, the values of which can be easily determined by direct measurement on a universal goniometer, and then no preliminary calculations are needed; using these same five angular values, using the basic formula, the determination of any face, not only measured, but and every possible one, given by the indices of the symbol, is produced, as mentioned, by simple additions and subtractions, but with the exception of the faces located in the original belt itself, that is, the belt of faces a and b. And in the textbook (“A Brief Course in Crystallography”) there is no formulas are given for calculating the angles between the faces in this belt. Here I will give the derivation of this remarkable and extremely simple formula.
The usefulness of compiling such detailed tables becomes clear, which, of course, will be compiled when it is necessary to make more accurate calculations of the densities of face networks or edge densities. In all cases in general, except for triclinic crystals, such calculations will be distinguished by significant simplicity, or more precisely, not at all will be needed when the tables are compiled. Accuracy can also be enhanced when using the graphical method, if we take as a basis for calculations not gnomostereographic (or gramstereographic), but gnomonic (or linear) projections.
Realizing the state of crystallography which I found 40 years ago, and comparing it with the present, I find that its transformation during this period is almost deeper than any other science. The closeness of crystallography and chemistry seems natural: both belong to sciences of an intermediate nature between exact sciences, the study of which is entirely reduced to the application of mathematical methods, and descriptive sciences, where the mathematical method is not applied at all. If we compare the role of mathematics in chemistry and crystallography 40 years ago and now, we will of course see that the latter has undergone a much greater transformation.
When dualism was recognized in the New Geometry, only two geometric systems were assumed: a system of points and a correlative system of planes. This note aims to show the special significance of such a particular case, which is expressed by the following theorem, if we call two such correlative systems, in which the spheroprims of one are correlative to the spheroprims of the other, cognate systems. All constructions, and therefore theorems, of one related system are transferred to another. In addition, I mean to show that it is possible to establish such systems that for each of its linear seconds it is possible to reproduce a related system of points on the plane.
In the article “Extreme simplification of zonal calculations and crystallographic calculations in general,” I noted the extreme simplicity that calculations of spherical bipolar coordinates obtain if we take Miller’s formula as a basis. Now I will supplement the formulas derived then with those that relate to the calculation of the cotangents of the angles formed with any initial face of the belt (see note). We can conclude that the system of zonal calculations with bipolar coordinates, in its special simplicity, is applicable without the use of a sequential, recursive course of calculation, and directly to faces with arbitrary complex indices.
Here I mean only what is meant by the word “twins”, and not those natural accretions that are caused by mechanical shifts - a case already discussed in my previous works. It seems to me that the experiments of the correct growth of dissimilar crystals on each other and the conclusions drawn from them by F. Barker completely establish the physical reason for the formation of twins. The most general conclusion from the principle is that the possibility of twinning growth along any plane of the complex, no matter how complex the symbol may be its expression, for cases of twinning faces with a complex symbol, is used in nature in exceptional circumstances.
Vl. Meyer addressed me with a letter, in which he reported on a graphical method of dividing a circle into equal parts using a simple technique (see article). Of course, theoretically this is not correct; it is impossible even to generally establish the perspective of points on a circle and on any straight line otherwise than by taking the center of the perspective rays on the circle itself. Mr. Meyer's method has no theoretical basis and can only be approximate in relation to the semicircle. This method can really be used at least in construction work, for example, when placing pillars and columns around the circumference of a circle.
When solving problems about the correct installation of crystals (as a basis for crystal chemical analysis), one constantly has to resort to symbol transformation, while verification of this solution is associated with the second and further transformations. There is a problem of expressing the final symbols in the original ones. Instead of the well-known conversion formula, in practice we use an abbreviated expression in the form of a determinant (see article).
The structure of such parts of the earth’s crust as the modern Ural Ridge is something so complex that no human imagination is able to comprehend it in all details, and any attempt in this direction comes down to a more or less detailed diagram. But especially grandiose mining areas hid within themselves not only this complexity of structure, but also so many subsequent changes and transformations that even a schematic representation of the processes taking place there encounters barely surmountable difficulties. It is enough to point out the intense activity of metamorphosis and weathering to note these features. See the results of the study in the article.
Since in general, to define a complex, four faces are necessary and sufficient, it is clear that with three given points, the vertices of any triangle, their poles constitute, further development of the complex is impossible, and it is absolutely necessary to know the position of some other fourth face. In general, the choice of such depends on our desire, and I will consider the case when these four points form the vertices of two adjacent triangles that have a common side.
A network for the hypohexagonal type was established and it was shown that the numerical law of the development of forms is essentially the same for this network, although now the faces are expressed by symbols not of three, but of four numbers; from these four you can always choose three, including the necessary the first number, which will be completely identical to the numbers of the first network. But even if we do not make such a selection, but limit ourselves to three numbers, of which one is in first place, and we choose the other two arbitrarily (that is, the second with the third , or the second with the fourth, or finally the third with the fourth), then the law in question here will still remain valid. Of particular importance is the distribution of even and odd numbers. This law is that of the seven symbols relating to any elementary triangle of the network, one certainly encloses three (that is, all), three encloses two, and three encloses one odd number; all other numbers of symbols are even.
There is nothing more natural than to generalize the conclusions of the previous note relating to three-digit numbers or a triangular geometric network to numbers of higher significance and, above all, to four-digit numbers, and the result is a tetrahedral network. Such a network of numbers has found its application for the chemical tetrahedron in petrography. Reflecting on the mathematical foundations of the construction of a triangular network, we will find that the main theorems remain valid for this network with a corresponding complication of the constructions themselves. This complication lies in the fact that the total number of points associated with one elementary tetrahedron (more precisely, a sphenoid) of any period no longer 7(3 + 3 + 1), but 15: four at the vertices, six midpoints of the edges, four midpoints of the faces and one midpoint of the tetrahedron itself.
In this article, I narrow my task to the highest degree and do not even mention the various geometric systems and their combinations that make it possible to solve it; I focus exclusively on considering one single system of the 4th stage on the plane, which most directly corresponds to the essence of the matter. This system is a direct extension of the system of parallel vectors, namely, it is expanded in the sense that vectors, as elements of the system, do not have to be taken as parallel. It is clear that through this condition the system increases by a step, that is, it becomes precisely a system of the 4th step (corresponding to the geometry of the space of 4 dimensions).
Кристаллы этого минерала из музея Горного Института были уже мною систематически описаны в специальной статье. В последнее время музей прибыли два новые интересные кристаллика этого минерала. Больше всего бросается в глаза необыкновенная их тонкость, доходящая до 0,1 м.м. при плоскостном размере, большем, чем квадр. сантиметр. На таких крайних разностях особенно поучительно ставить вопрос о существовании зависимости между формою и комбинацией.
А. Э. Купффер, раздробляя породу из San Zenito в Калифорнии, в изобили и содержащую почти черные и довольно крупные кристаллы нептунита, выделил между прочим превосходно образованный кристалл довольно густого синего цвета, развитый в комбинациях типичного ромбического кристалла с четырьмя весьма острыми пирамидами (см. статью).
Воспользовавшись имевшимися у меня полушариями калистых квасцов и соответственными углублениями в крупном кристалле я пожелал испытать, как при той же кристаллизации влияют примеси в растворе, не оказывающие разлагающего действия на вещество, заключающееся в растворе. Как вероятный вывод—образование граней, хотя и плохого достоинства, с более сложными символами на сферических поверхностях, но столь малой величины, что вообще рефлексы не уловимыми, и начинают становиться уловимыми только некоторые из них, благодаря примесям, улучшающим кристаллизацию.
Вициналоидами или вицинальными поверхностями называются те поверхности, который, составляя истинные грани кристаллов и весьма приближаясь к плоскостям, на самом деле не есть плоскости, а весьма сложные и разнообразные кривые поверхности. Но если вообще при росте кристалла скучивание, то есть непараллельное положение частиц друг на друга происходит хаотически то есть одинаково во всех направлениях (полная нестройность скучивания), то нельзя отрицать возможности существования причин, нарушающих эту полную нестройность и производящих неполную нестройность.
В основе современного представления о законности в образовании граней лежит положение о согласии порядка важности граней (проявляющейся как в частности их появления, так и в их величине) с порядком их ретикулярной плотности. Это положено выведено на опыте как закон статистического характера то есть не как закон точный, проявлявшийся всегда и безусловно, а как законность, проявляющаяся в значительном болышинстве случаев. Исключения, который мы вообще находим на опыте, отнюдь не исключают мысли об абсолютном значении порядка плотности граней; но они указывают, что на образование граней, кроме этого абсолютного фактора, влияют и другие, значение которых еще не удается выразить численным способом; и эти факторы могут быть довольно многочисленны, так как на степень образования тех или других граней влияют и разные внешние, отчасти трудно уловимые условия.
Отмечу не только важное для кристаллохимическаго анализа разнообразие, усматриваемое из этих таблиц, но и важное значение того промежуточного минимума, который замечен для главных чисел. Нужно заметить, что в изотропных комплексах различие между тетрагоналоидными и тригоналоидными пропадает и вообще кристаллы могли бы быть отнесены к псевдокубическим. Но если присоединить к ним еще настоящие кубические кристаллы, то мы именно для этого промежутка получили бы особое скопление кристаллов, что для анализа явилось бы фактором неблагоприятным, и вот оказывается, что как раз в этом промежутке естественно получается некоторое разрежение в распределении.
Амфибол. Пироксен. Эгирин. Энстатит. Гиперстен. Бабингтонит. Лиеврит (ильваит). Берилл. Фенакит. Трустит. Виллемит. Сфен (титанит). Паризит. Золото. Церуссит. Арагонит. Кварц. Оливин. Нептунит. Киноварь. Целестин. Барит. Подробное описание кристаллов см. в статье.
В записках Г. И. на стр. 259 по поводу составления таблиц для кристаллохимического анализа уже были приведены некоторые статистические данные по этому предмету, хотя эти данные и основывались на меньшем материале, чем имеется сейчас в моем распоряжении. Сейчас сделан крупный шаг по составлению этих таблиц, а именно составлено 3730 диаграмм по материалу, заключающемуся в 42 томах Zeiteschrift für Krystallographie как в его оригинальных статьях, так и в рефератах (несколько сот этих диаграмм уже проверены сотрудниками и таким образом для них установки, а следовательно и место в таблицах, закреплено).
В настоящее время кристаллография широко пользуется графическими приемами для решения своих задач. В числе целей, ставимых при развитии графических операций, преследуется также и увеличение точности. Что касается конструкции нового прибора, главным образом отметим гораздо большие размеры прибора, коего черное полушарие, как основная рабочая часть прибора имеет диаметр аршин с небольшим. Второй прибор есть универсальный прикасательный гониометр с тремя осями. Он служит для обыкновенного измерения по универсальному методу столь крупных кристаллов, что они уже не могут быть укреплены на кристаллоносце обыкновенных гониометров. Третий прибор предназначен для облегчения процесса кристаллизации. Принцип работы состоит в ритмическом нагревании и охлаждении сосуда с раствором, в котором происходит кристаллизация.
We now know that geometric systems can be very numerous and varied, since very diverse geometric images can be taken as elements of systems. To establish any such system, it is necessary to determine the complete set of its elements and provide proof that from two elements arbitrarily taken from it it is possible to unambiguously compose an infinite set of them such that, by replacing the two taken in it with two arbitrary other elements included in its composition, we from them they would also unambiguously deduce the same set, which would constitute the linear principle of the system.
The author concludes that there is an unlimited number of geometric systems of the same level, deduced from each given one. Since the conclusion about the possibility of reproducing from any given system another, twin one, is not limited by any conditions and is determined by the possibility of the same positional constructions as for all systems, it is clear that it is equally applicable to twin systems. In other words, we can reproduce a new, twin system not only from any geometric systems in general, but on absolutely the same grounds and from each twin system.
Among the remarkable examples of potassium feldspar in the Museum of the Mining Institute there is a very large adularia quadruple from St. Gotthard (Fibia), reproduced in Fig. 1 of the attached table (up to two decimeters in length).
The author came across clear signs (Fig. 3) of the formation of stripes in the microcline in thin section from the shores of the White Sea (No. 8, that is, from Gorely Island in the Keretsky roadstead). I felt obliged to present such an image, perfectly clear at 120x magnification.
Since the plane P is only a special case of a curved surface of the 2nd order (conosecond) K, then from this and some other conosecond, given completely arbitrarily, their linear primacy is quite unambiguously determined. Such will be the aggregate for the determination of which these data are sufficient and can be replaced by any two conoseconds of the same aggregate.
Correlativity is established not only between a system of points and a system of planes, but also between transformations of these systems. It is precisely because of correlativity that this theorem has a dual meaning, so that in its formulation an ordinary lattice can be replaced by a polar one and vice versa. The author considers it necessary to publish this theorem in view of the fact that in crystallography, to determine the symbol of a complex, we precisely perform the operation of shifting the polar lattice (using the gnomosteographic projection), while the essence of the change that the ordinary lattice undergoes in this case remained unknown.
I began experiments of this kind back in 1901, when I cut out circles or small rings on thin sections of rock salt and alum that separated the inner convex from the outer concave spherical line; I put a drop of an unsaturated solution into this annular space and covered it with a coverslip, which I sealed with Canada balsam.
In the article “Precise imagery of space points on a plane”, the problem of such an image in three different elements is solved: vectorial and ordinary circles, and in parallel vectors. The practical application of images in parallel vectors of a system of mines is also given there. I will now show an essential application of the theory to the representation by vectorial circles of the spatial lattices of each structurally studied crystal.
We know that from two given points e with e' and a conical section K on the plane, we can reproduce a curved surface of the 2nd order, if from these points we take one e as the center of the second of the rays, and the second e' as the center of the second of the planes and bring these two seconds into a correlative relation so that the ray ea (a point on the conic section plane) will be considered correlative to the plane e'A, where A is the polar of the point a with respect to the conic section K. It is known that in such a surface a set of rays and their correlative planes.
No matter how elegant the construction of conic sections using Pascal's theorem is, it does not have sufficient generality, since it is applicable only for five real points of a curve, and in practical application it is more difficult than some other methods.
I consider it useful to note one property of stereographic projection, which it does not open up new ways for solving problems, still can contribute to greater accuracy in solving some of them.
Among the deviations from the laws characteristic of real (ideal), that is, completely crystalline-homogeneous individuals, small deviations are often noticed both in the position of the edges and in general in the intergrowth of subindividuals.
By this general name we mean all those curves and surfaces that are represented in the geometry of harmonic segments. Indeed, in this geometry, every midpoint of a segment uniquely corresponds to both end points of this segment; consequently, whatever curve is represented in this geometry as a set of end points of segments, its points are always grouped into pairs that are harmonic with respect to the main orthogonal sphere O, and the end points are located on the radius of this sphere.
One of these systems or spherical geometries can be considered generally known, although I am not aware that anyone has categorically noted its complete parallelism or equality with the geometry of points on the plane. This geometry on the sphere introduces one limiting condition that it deals only with points on the sphere. Therefore, although its points occupy all three dimensions of space, it is essentially the same geometry of two dimensions (that is, the second stage), as is the corresponding geometry on a plane, in which the limitation is to consider only points on one plane. Despite a long line of first-class researchers of this mineral, starting with Brooke, who studied this mineral in 1824, its crystallization, actually the correct installation of its crystals, has up to now encountered significant difficulties. I can now make use for this purpose of an excellent specimen from Alston Moor, in the Museum of the Mining Institute, from which hundreds of crystals suitable for this purpose could be extracted. I extracted 15 crystals and subjected them to continuous measurement on a universal goniometer.
Despite a long line of first-class researchers of this mineral, starting with Brooke, who studied this mineral in 1824, its crystallization, actually the correct installation of its crystals, has hitherto encountered significant difficulties. I can now make use for this purpose of an excellent specimen from Alston Moor, in the Museum of the Mining Institute, from which hundreds of crystals suitable for this purpose could be extracted. I extracted 15 crystals and subjected them to continuous measurement on a universal goniometer.
During the radical revision and putting in order of the minerals of the Museum of the Mining Institute, which is now being carried out by A.E. Kupffer, I got an opportunity to get acquainted in detail with the figures mentioned in the title, which are sometimes developed with great clarity on a fairly large number of specimens from various deposits of this interesting mineral, so abundantly represented in the museum.
Natural crystals are the most difficult object for the method of crystal-chemical analysis, and I must admit that my attempt to give the correct setting to all minerals turned out to be imperfect in many points. But it was precisely this imperfection that prompted the further development of criteria for correct installation, and pointed to the need to reconsider and individually study the crystals of many minerals.
Having specially studied pyroxenes, I came across, among other things, intergrown crystals resembling twin intergrowths. Let me present here the results of one such study, based on the numerical data of which the attached diagram was compiled.
The criteria used so far contained the main, fundamental drawback, which consisted in calculating the grid density of each given complex as an isotropic complex. Although the criterion with this drawback was used quite consciously, for the sake of simplicity, in view of the complexity of the operation of calculating the density of the grids, but, of course, there was always a desire to eliminate it, if only a way could be found to determine this density quite correctly, without resorting to simplifying, but still an erroneous assumption.
I will provide the additional theorems on linear collections of the geometry of vectorial spheres and quadratic collections of ordinary spheres. For a detailed description, as well as a comparison of megaspheres of vectorial and ordinary spheres and an overview of the system of parallel vectors, see the article.
If you have collineations of two systems, then both systems are equal, because both are in the position of involution, and any point a of the system is collinear to the same point a of the collinear system, regardless of which of these two systems the given point belongs to. But now let’s replace one of the systems with a system similar to it, and take the combined centers of collineation of both systems as the similarity center E. It is clear that under this condition the systems can no longer be brought into the position of involution, and therefore the construction of homologous (collinear) points becomes more complicated, and in any case, for each given point in the collection we will obtain two different homologous points, depending on which of the systems this point applies.
We outline the possibility of a path for the derivation of an indefinite number of new geometric systems. The main feature of the problems of New Geometry is the indefinite multiplicity in the application of theorems, in contrast to the individuality of conditions in the formulation of problems with which ancient and analytical geometry deals. Problems of a metric nature, for this reason, are not at all within the scope of this discipline; but it would be inaccurate to say that it includes only problems solved by positional constructions (for which reason New Geometry is more often called positional or projective geometry).
Mining engineer Natsvalov kindly delivered several samples of sublimate crusts during firing of kupfermatte at the Kedabek plant. The specimen was of interest because of its beautifully formed, although mostly in the form of skeletons and growth figures, octahedrons with small blunt cube faces.
This difference is manifested in the movement of the dissolved substance in the layer of solution separating the different faces of the crystal of the substance from which the solution is obtained. If there were any doubt about the similarity in this regard of the properties of a natural edge and, for example, a plane of cleavage parallel to it, then the mentioned technique provides a means of resolving it.
First of all, interest was aroused by the huge Bavinsky twin of orthoclase from the vicinity of Kyakhta, depicted in Fig. 1, and not only by the clarity of its formation, but also by the beautiful almandive crystals included in it. A special optical study carried out by V.I. Sokolov showed that the orthoclase substance is germinated by plapoclase around No. 5, representing well-formed polysynthetic twins according to the albite law, and one individual of the twin is concordant with the orthoclase substance.
Whoever sought to give himself a clear account of the chemical relations of rocks had to see with amazing clarity all the imperfections of our information on this issue. We have to admit that real, exact science has hardly yet touched upon this important issue, and the most primitive empiricism reigns here. A large number of types of rocks are exhibited, receiving numerous names, but there is not even a generally accepted criterion for distinguishing one type, or, more precisely, a type of rock from another.
If there is a parallelism between the theorems of geometry of the points in space and the circles on the plane, then the idea of the possibility of accurately representing points (and, consequently, images obtained from points of other images) of space by circles on the plane naturally arises. The purpose of this article is to develop the most perfect and simple way of depicting such an image. However, in addition this article introduces two new geometric systems that are also used for the same purpose, namely, the system of vectorial circles and the system of vectorial segments or simply vectors.
The formula in question here is a logical consequence of two already known formulas, which were given in full by the author of the doctrine of symmetry, namely in the part that was published under the title “Symmetry of Finite Figures.” The formula, applicable to any group of symmetry axes (type of alignment symmetry), but of course not applicable to one axis taken separately, makes it possible to directly derive the value of symmetry from the number of symmetry axes. From it, by the way, it follows that the value of the symmetry of the combination is certainly even (which is understandable, in view of the obligatory presence of double axes of symmetry in the aggregates), and therefore the value of the symmetry of those types where, in addition to the axes of symmetry, the elements of direct symmetry are also included, is certainly divided by four.
A.E. Kupfer brought from a well-known mercury deposit pieces of light burnt (and therefore reddened) sandstone, covered with thin crystalline crusts. The crystals are thin-columnar, completely colorless, with a diamond luster and in general look very similar to calomel. However, the on-site analysis, having stated the presence of chlorine and mercury, gave an insufficient amount of chlorine, which is why it was assumed that these were not calomel, but crystals of some other mercury chloride. In these cases, the decisive methods are the usual methods of crystal-chemical analysis reduced to measuring the crystals and their correct installation
Although at present there is hardly any disagreement between specialists on the issue indicated in the title, it still seems impossible to point to direct experience that imediately resolves this issue beyond any doubt. Concerned about this form of experience, I focused on such a simple and convincing one that I can consider the goal achieved. According to the idea that was developed in my article “Observations and Experiments on Crystallogenesis,” solubility in relation to each face of a crystalline substance is directly proportional to its network density, and therefore, the more complex the symbol of the observed face (with the correct installation of the crystal), the smaller it is, and for irrational faces it is even equal to zero.
