Rolls Series

“ Fig. 81.
The use of a roll in finishing dates from the end of the fifteenth century, and some satisfactory bindings were decorated with its aid. The ease with which it can be used has led in modern times to its abuse, and I hardly know of a single instance of a modern binding on which rolls have been used for the decoration with satisfactory results. The gain in time and trouble is at the expense of freedom and life in the [191] design; and for extra binding it is better to build up a pattern out of small tools of simple design, which can be arranged in endless variety, than to use rolls. ”

“ In addition to the eight rolls of paper already mentioned eight other rolls are in position, so that when any of the rolls run out the roll-carrier may be turned on a turntable, and the new roll of paper quickly pasted to the end of the depleted roll. ”

“ You ought to make a very small flame, if you leave the top of the tumbler open."
"How can we make a small flame?" said Rollo.
"One good way," replied Miss Mary, "is to roll up some paper into a very small roll. I will show you how."
So Miss Mary took a piece of paper, and cut it into the proper shape with her scissors, and then rolled it up into a long and very slender roll; one end of it was not much larger than a large knitting-needle. She gave this to Rollo, and told him that, if he tried the experiment again, he must light the small [140] end, and it would make a flame not so big as a pea. ”

“ His mother and James looked up, wondering what Rollo meant. He was looking out at the side of the carryall, at something about the wheel.
“What is it,” said his mother.
“Why, here is a large fly trying to light on the wheel, and every time his legs touch it, it knocks them away. See! See!”
“Yes, but you must not attend to him now. You must listen to my lecture. You promised to give your attention to me.”
So James and Rollo turned away from the window, and began to listen again.
“I have told you now,” said she, “of one kind of playthings—those that give pleasure from their novelty only. ”

“ In Initial Series the count is always forward, and, in general throughout the inscriptions. The very few cases in which the count is backward, are confined chiefly to Secondary Series, and it is in dealing with this kind of series that the student will find the greatest number of exceptions to the general rule. ”

“ You can write on the under side of a lilac leaf with the point of a pin.”
“Can we?” said Lucy, with a look of curiosity and pleasure.
“Yes,” said Mary, “you need not write the whole name. You can write the first letter—that will be enough. R. stands for Rollo, L. for Lucy, H. for Henry, J. for James, and N. for Nathan.”
“Ho!” said Rollo, “Nathan won’t do for a regent.”
“Perhaps not,” said Mary; “each one of you must vote for the one you think best. Now get your lilac leaves, and I will drop you down some pins.” ”

“ If a work was divided into several volumes (see below) , the rolls were put together in a bundle (fascis) and kept in a wooden box (capsa, scrīnium) like a modern hat box. When the cover was removed the titulī were visible and the roll desired could be taken without disturbing the others (Fig. 190) . The rolls were kept sometimes in cupboards (armāria, §231) , laid lengthwise on the shelves with the titulī to the front, as shown in the figure in the next paragraph. ”

“ We may omit, for the time, from our notice, the before-mentioned possibility of a knowledge of the world that would reduce it to a single serial order of the general type of the Well-Ordered Series of whole numbers, where every term has one coming next after it. If we abstract from that possibility, we are left to the conception that Between any two facts there are to be found various series of intermediaries of the type now defined. The world thus regarded will consist for us of all these interwoven series, and will constitute a single System. ”

“ George said nothing, and so Rollo went away to find the landlord.
That night, just before bed time, Mr. George asked Rollo what time he was going to set out the next morning.
"Immediately after breakfast," said Rollo.
"Are we going to ride or walk?" asked Mr. George. [Pg 201]
"We are going to walk over the pass," said Rollo. "The road is too steep and rocky for horses. But then we are going to have a horse to carry the trunk."
"Can you put our trunk on a horse?" asked Mr. George.
"Yes," replied Rollo, "the guide says he can." ”

“ George.
"Because the guide book says there will," replied Rollo. "They always have boats there to take people that come along this road to Lucerne."
"Why do they not go all the way by land?" asked Mr. George.
"Because," said Rollo, "the whole country there is so full of mountains that there is no place for a road."
Just at this time the guide got up from his seat, and seemed ready to set out upon his journey; and so Mr. George and Rollo rose and went on. ”

“ You had better place the card in such a position, too, that its sides shall lie in the direction of [Pg 21] north and south. Then the shadow at noon will lie along exactly in the middle of it. You must get a large and stout pin, too; and drive it in firmly, a little way, with a small hammer. It will be well, too, to drive another smaller pin into the other end of the card, so as to keep it fixed in its north and south position.”
“How can we know when it is north and south, exactly?” said Lucy.
“You cannot do it exactly,” said Rollo’s mother; “but you can get it pretty near. ”

“ There were men, some dressed like gentlemen, and others wearing blue, cartmen's frocks; and women, some with bonnets and some with caps; and children of all ages and sizes; and soldiers without number, with blue coats, and dark-red trousers, and funny caps, without any brim, except the visor. In the midst of all these multitudes Mr. George and the gentleman who was with him slowly led the way up the side avenue, Rollo and Jennie following them, quite bewildered with the extraordinary spectacles which were continually presenting themselves to view on every hand. ”

“ This certainly would seem to indicate that an Initial Series is to follow. Moreover, the fourth glyph of the eight-term number following in A2-A9, inclusive (that is, A5) , records "Cycle 9," the cycle in which practically all Initial-series dates fall. Indeed, if A2, A3, and A4 were omitted and A5, A6, A7, A8, and A9 were recorded immediately after A1, the record would be that of a regular Initial-series number (9.3.6.2.0) . Can this be a matter of chance? If not, what effect can A2, A3, and A4 have on the Initial-series date in A1, A5-A9? ”

“ Rollo stood silent and thoughtful as Jonas said this, with his piece of wood in one hand, and his hatchet in the other.
“It takes a great deal of patience to make a thing which we never made before.”
“Why?” said Rollo.
“O, because there are always unforeseen difficulties. We don’t know exactly how to do it, and are apt to make mistakes; and so we spoil some of our work, and this makes us impatient and fretful.”
“But I could not help coming to you,” said Rollo, “when I found I could not sharpen my wedge.” ”

“ But now he said to Rollo,
“I could let you have all the books of prints and engravings I have got, and see them all at one time, and that would be giving you Aladdin’s lamp, exactly, so far as my pictures are concerned.”
“Well,” said Rollo, clapping his hands.
“But then, in a short time, you would get tired of looking at them; you would become satiated, and would in fact spoil the whole pleasure by attempting to enjoy it too fast. But then I think it would perhaps do you good.” ”

“ The actual construction of the series would be a typical instance of a self-defeating and therefore internally contradictory purpose, inasmuch as it would involve the actual completion of an unending process. Hence we seem forced to make a distinction which Prof. Royce has perhaps unduly neglected. If your purpose of ordering the number series on a definite plan means no more than the formulation of a rule for obtaining any required number of terms of the successive series, it can be executed, but does not involve the indefinite regress ”

“ I can multiply it by ten. Ten times nothing are nothing. So I can divide it. Five in nothing no times, and nothing over."
Rollo was somewhat perplexed by this argument, and he did not know what to reply. Still he would not admit that nothing was a number—still less that it was an odd number. He did not believe, he said, that it was any number at all. The boys continued the discussion [A] for some time, and then they concluded to go and refer it to Mr. George.
And here I ought to say that Waldron had an artful design in taking nothing in his hand, when he called upon Rollo to say, odd or even. ”

“ He and Rollo went out into the garden to look at the establishment, and found every thing as they had left it the night before. Rollo felt quite confident of the success of his experiment. The only thing that gave him any uneasiness was the want of a queen bee. He and Henry were just speculating upon the expediency [Pg 119] of sending in a bumble-bee instead, for a king, when their attention was arrested by hearing Jonas calling Rollo. They looked up, and saw him standing at the garden gate. ”

“ Rollo ran into the house with great haste and eagerness, leaving all the doors open, and calling out, “Father, father,” as soon as he had begun to open the parlor door.
“Father, father,” said he, running up to him, “I wish you would let Jonas go with us to-morrow.”
Now, Rollo’s father had come home but a short time before, and was just seated quietly in his arm-chair, reading a newspaper, and Rollo came up to him, pulling down the paper with his hands, and looking up into his father’s face, so as to stop his reading at once. Heedless boys very often come to ask favors in this way. ”

“ Rollo presented the appearance of a house from which the inhabitants are meditating an immediate journey with all their effects. Packages of all sizes and descriptions had accumulated, to a number which became intrusive upon the notice of said inhabitants.
'What shall we do to make a clearance?' Rollo had said, laughing, as his eyes went round the parlour. 'I wish, Hazel, you would look at these things, and see what use you can find for them. Take Byrom to open packages and do them up again, and let him ticket them according to your orders. Will you? and when I come home I will help. ”

“ So Rollo would not go upon the bridge.
"Then let us go out on the forecastle," said Waldron. He pointed, as he spoke, to the forecastle, which is a small raised deck at the bows of a steamer, where there is an excellent place to see.
"No," said Rollo, "I will not go on the forecastle [Pg 19] either. Uncle George's rule for me on board ship is, that I may go where I see other gentlemanly passengers go, and nowhere else. The passengers do not go on the forecastle."
"Yes," said Waldron, "there are some there now."
"There is only one," said Rollo, "and he has no business there." ”

“ In Case II the infinite series is said to be nonconvergent [3] . Series under this head may be divided into two classes.
First Class. Divergent series, in which the sum of terms increases indefinitely in numerical value as increases without limit; take for example the series
. As increases without limit, increases without limit and therefore the series is divergent.
Second Class. Oscillating series, of which
is an example. ”

“ Henry and Rollo looked upon him with an expression of ludicrous gravity and perplexity.
“What are you laughing at?” said Rollo.
Jonas could hardly control himself sufficiently to speak; but presently he succeeded in asking Rollo if he supposed that bees would make honey there.
“Certainly I do,” said Rollo, with a positive air. “Why should they not? They don’t care what shape their hive is, or what it is made of, and this flower-pot is as good as any thing else. There! there! see, Henry,” he exclaimed, interrupting himself, [Pg 121] and pointing down to the flower-pot, “one is coming out.” ”

“ There were cabinets inlaid with silver and pearl, and elegant cameos and mosaics, and a profusion of other such articles, all of which Rollo had very little time to examine, as the schloss-vogt led the party forward from one room to another without much delay.
The rooms themselves, in respect to form and arrangement, were almost as curious as the articles which they contained. Every one seemed different from the rest. You were constantly coming into the strangest and most unexpected places. ”

“ The boy had reached the top of the staircase when Rollo went out, and was just then coming along the hall. He looked at Rollo with a smile as he came toward him, and this encouraged Rollo to speak to him.
"Can't you find any one to play with you?" said Rollo. ”

“ By this principle it appears, that all tables in which each series of differences continually increases, may be produced by the operation of addition alone; provided the first terms of the table, and of each series of differences, be given in the first instance. But it sometimes happens, that while the table continually increases, one or more serieses of differences may continually diminish. In this case, the series of differences are found by subtracting each term of the series, not from that which follows, but from that which precedes it ”

“ For instance, a passenger on board a packet ship once asked a sailor what time they would heave the log. 'The log,' said the sailor, 'they always heave the log at nine bells. When you hear nine bells strike, go aft, and you'll see them.' So the passenger watched and counted the bells every time they struck, all the morning, in the hopes to hear nine bells; whereas they never strike more than eight bells. It was as if a man had said, on land, that such or such a thing would happen at thirteen o'clock."
Rollo and Jennie laughed. ”