studiot

As I said it it part of a larger scheme. Here is your question fully worked out (first 5 pages of Bonolo Non Euclidian Geometry)

(as KJW says) The deduction is:- If the sum of alpha and beta is not less than two right angles then l and m do not meet on the same side of the line as alpha and beta. There are three cases to consider, alpha plus beta < , > or = two right angles. Euclid does not start out assuming there are any lines in the plane that do not meet, that is his deduction shown in very short form by Stillwell. Heath's original translation goes into greater detail and even offers the original Greek.

The deduction is only valid if and only if the axiom is valid. Remember the axiom (they called it a proposition) is stated without proof.

The version of theaxiom you state is Euclid's own original. Even in his time there were several altbernative versions (eg Proclus, Arisole) aand much debate about he subject. What should be remembered is that Euclid built a coherent structure for Geometry and offered his version in line with the posiion in that buildup postulate 5 occurs. All the properties of parallel lines (apart from not meeting) are deduced by triangles after this raher as in your 'deducion' paragraph. What is your interest in his ? Many books have been written about this subject.

I had always been led to believe that US prices were well below UK prices but reading this BBC articlwe makes me wonder. BBC News'I had no electricity for six months': American families...Rising electricity costs have emerged as a key cost-of-living concern, pushing families further into debt. Over £1300 a month for electricity and rising ?

Part of this series of DVD lectures shows Dr Stewart in Western Australia picking up meteorites amongst some of the oldest rocks on earth at the surface ie billions of years old.

Z are the integers, not the natural numbers, N. "Find a natural number..."

Not quite sure about this since surely every number in N can be wriiten as the sum of 3 squares in R. For instance 14 = (3)2 + (√3.5)2 + (√1.5)2 and 28 = 14*2

No it is not a silly question but then it is also not a simple question (looks can be deceptive) . I did not answer before partly because of this and partly because a lot of things have been happening in my world just lately, not least including computer failure. The questions are actually quite deep and you would and a person would only need the answers if he was going to study the subject more widely. It would also require some preliminary work, but since you have come back and asked again perhaops you are up for that ? A good place to start would be to compare the normal distribution function with one half cycle of a sine function as they seem so similar at first glance. But they are actually quite different. The sine function is cyclic (important new term) The normal dist function is not cyclic. (exchemist is not quite right as the 'tails' stretch from negative infinity to positive infinity though it is almost never used in that way)

😀+1 Joking apart, I think this mechanical/chemical reason is responsible for many of today's dietary induced woes. Full Fibre has nothing to do with broadband and everything to do a healthy diet, for sound evolutionary reasons.

Still no sign of an answer to my question.

Welcome back @Mordred Season's Greetings.

The short answer is that no one actually knows. New discoveries happen all the time and the whole subject is under continual review. BBC NewsDark Energy may be changing and with it the fate of the U...A mysterious force called Dark Energy might be changing, in a way that challenges our current understanding of the nature of time and space.

It is the hallmark of false claims that the claimant ignores or fails to answer simple polite questions such as my question Is your machine silent in operation ?

I didn't see an answer to my very simple question. Is your machine silent ?

Pity you didn't answer my question so here is another one now that you have post 1/4 of a description. It is ironic that this claim involves, at least in part, Joule's original lead shot experiment with energy. Anyway Is your machine totally silent ? Nothing falls without making some noise and, as Joule found out, without generating some heat.

Difficult decision whether to post this in Phi's thread or swansont's thread.

If you can't be bothered to read the site rules here about posting before you post anything here then why should I be bothered to read anything you post here or anywhere else ?

First what does this mathematical nonsense have to do with Classical Physics ? Mathematically the issue of a next number depends upon which number system you are referring to. Also the Axiom of Choice guarantees a well ordering of most number systems. What you are trying to explore is that numbers in some numbers systems have no nearest neighbour. To study this you need to look at neighberhoods, completeness and density from a mathematical set point of view.

I don't think you quite mean this so perhaps youd like to elaborate a bit about what you really mean.

Intriguing comment. +1 It is not conventional, but I suppose since one way to generate the natural number system is from 'nothing' , that is the empty set, a case could be made for this. However the empty set is not the number zero and any placement of zero in a number system causes controversy as it fits some applications of that number system but not others.

Thanks for that folks. I'm not sure about theatrical Toksvig, but I appreciate the answers. I think they are consistent with my understanding that it is about the direction of observation and best explained to someone who has at least the most basic Tech Drawing.

It is often stated that plane mirrors invert left to right but not top to bottom. It is given as the reason why the word AMBULANCE is often written on the front in 'mirror writing'. Yet if you are followed by that same ambulance and it indicated to turn right or left you do not see the indicators switched. So if the ambulance indicated left behind you you see the left hand indicator flash in your mirror. If anyone has a clear predictive explanation of images in plane mirrors I would be grateful for them to post it.

I think a point to consider is What happens when everyone along the Rhine wans to do this ? I have a doctor friend who lives along the Rhine and already has a private deep bore water source heatpump.

Taking regard of the above points I would perhaps reconsider my comments since whilst I agree that many, if not most, measurements are made with reference to a standard, not all measurements are actually numeric. And even when the output is a 'number' it maynot be a cardinal or ordinal but a number used as a label. The Richter scale, the Beaufort scale the Mohr scale, the Atterberg scale. Lustre, Sheen come to mind. So regarding measurements as injections is perhaps a bit over the top and certainly specefication of the set from which the numbers are drawn.