By now, I imagine you've figured out that this story has a kind of "false bottom," like a jewelry box. But there's something interesting even under the second bottom. Let's allow for non-integer values of k. Then what value of k will make Terentius's reward the greatest? It's clear enough that for this k the mass of the coin he takes on the 10,000th day must be exactly 700 kg—that is, it must have a denomination of 140,000 brasses, which means that k- l^OOO1/"99 = 1.0012.
Then Terentius's total income over more than 25 years of his daily visits to the treasury will come to S = (k10000 - l)/[k - 1) = 120 million brasses! This is many times more than the sum he requested of the emperor. In truth, the real avalanche doesn't come crashing down—it just creeps along. So here is how I would advise Terentius to respond to the emperor's seemingly attractive offer:
"Sire! Such a reward is too generous for me. Not only that, it will lessen the treasury so rapidly that severe damage will he inflicted on you and on the entire state. So I can't agree to such a sharp growth in the coins' value. But it would be impudent of me to turn your offer down completely. Might I ask only one thing of you: let the value of the coins grow, but not so rapidly. I'd be completely satisfied if each coin would be more massive than the previous by twelve hundredths of a percent." (Note: most probably they didn't know percentages at that time. I imagine, though, that Terentius could have expressed his wish in some other way.)
Nothing ventured, nothing gained. Maybe the emperor would have swallowed the bait without noticing the hook—which would eventually lead to the bankruptcy of the empire.
Actually, in this case a certain difficulty arises: the values of the coins won't be expressed as integers, which probably wasn't allowed at that time. No matter—Terentius could propose a magnanimous correction: rounding down to the nearest integer! This wouldn't cost him too much, because the damage will definitely be less than ten thousand brasses, which is nothing compared to his income.
Of course, it's easy for us to solve the financial problems of the brave general. But how would Terentius himself respond to my advice? It's not unlikely that he would find the proposal strange, to say the least. After all, he would have to wait 20 long years for the bulk of his reward. In the first five years Terentius would receive less than 6,600 brasses, and during the first year and a half he'd have to come every day for a one-brass coin! So who of the three is the strangest: the emperor, the general, or I? It's up to you to decide. At any rate, I can't help wondering what Yakov Perelman would have thought of this interpretation of his story. I'd like to think he would have been amused.
5.1.2. Slips of the Tongue
By Michael T.Motley
Several years ago, in the course of being interviewed for a job, I was introduced to a competitor for the position. Extending my hand and meaning to say, "Pleased to meet you," I accidentally said, "Pleased to beat you.” Although both of us laugh about it now (neither of us got the job), the slip made for considerable embarrassment at the time.
What caused the slip? Almost a century ago Sigmund Freud asserted that hidden meanings could be read into all verbal slips. More particularly, he held that all slips of the tongue reveal the speaker’s hidden anxieties and motives. Among those who study the cognitive processes responsible for language and speech production, that hypothesis has long been unpopular. The "Freudian slip" is difficult to examine in the laboratory, and is neglected in favor of hypotheses that were easier to test. Moreover, theorists tended to view the production of speech as a more or less autonomous process - one with no room for involuntary influences by motives, anxieties or other factors irrelevant to the speaker's intended message.
Further still, the categorical nature of Freud’s claim that all slips have hidden meanings makes it rather unattractive.
Serious attention to these questions and related ones has been renewed in the past decade. Paradoxically, however, the modern interest in verbal slip derives from an. interest in error-free speech production. Spoken language is among the most complex and mysterious behaviours in the human repertory. It is one of the dwindling numbers of behaviours that continue to resist computer simulation. 'The fact that speech production is usually error-free makes the process even more remarkable. At the very least a speaker wanting to express a thought must choose words that fit the intended meaning, must select grammatically legitimate organization of the words and must supply the appropriate motor commands to the larynx, the tongue and the lips. All these decisions and formulations of signals can occur in an instant, and so it is unlikely that they are under conscious governance. On the other hand, what people say is so often original - as a sequence of words if not as a thought - that speech is unlikely to be the result of mere reflexes.
The complexity and efficiency of speech production make it difficult to study: its constituent decisions and operations follow one another too quickly to be easily isolated and examined. It is therefore a good thing (at least for students of human speech) that people make verbal slips. In effect, a slip of the tongue offers freeze-frame observation of the speech-production process. When someone says "magician" instead of "musician" the mistake affords a glimpse into the speaker's word-selection process. When phonemes are transposed ("tea and flick spray" instead of "flea and tick spray") or substituted ("brouse" instead of "blouse"), other windows are briefly opened.
5.1.3. Archimedes and Lever
Archimedes was the first to develop a logical proof for the principle of the lever but, unfortunately, all his experiments were made with a straight lever in a horizontal position.His statements apply only to this special case and cannot be considered as a proof of the general principle of the lever. In his discussion of the lever he gives the following axioms and explanations:
(1) Weights which balance at equal distances are equal. For if they are unequal, take away from the greater the difference between the two; the remainders will then balance, which is absurd. Therefore, the weights cannot be unequal.
(2) Unequal weights at equal distances will not balance but incline toward the greater weight. For, take away from the greater the difference between the two; the equal remainders will therefore balance, and if we add the difference again the weights will not balance but incline toward the greater.
He also demonstrates by similar logic that unequal weights will balance at unequal distances, the greater weight being at the lesser distance.
Archimedes finally concluded that bodies attached to a lever are in equilibrium when their distances from the point of support of the lever are inversely proportional to the weights. As has been previously pointed out,these conclusions were all drawn from investigations made with straight horizontal levers, that is, the forces were always perpendicular to the lever arm.
From these discussions of the action of the lever he proceeds to the problem of the center of gravity. He shows that for two equal weights the center of gravity is at the mid-point of the line joining the centers of gravity of the individual weights. He then proceeds to show that if three equal magnitudes have their centers of gravity in a straight line at equal distances apart,the center of gravity of the three weights or of the entire system will be identical with the center of gravity of the middle weight.
He defines center of gravity in the following manner:" In every heavy body there is a definite point called the center of gravity in which point we may suppose the weight of the body collected".