What's the origin of the idiom "cut corners"?

1869: OED

The OED has:

to cut a corner or corners : to pass round a corner or corners as closely as possible; fig., to pursue an economical or easy but hazardous course of action; to act in an unorthodox manner to save time; also, to act illegally.

Their first quotation is 1869 (see ghoppe's answer).

1863: Longer shortcuts

An article entitled 'About "Going Straight On"' in The Oxford Magazine and Church Advocate (Vol. III., October 1863, No. 36, page 340) warns us of the pitfalls of cutting corners:

I do not believe, either, in what we used to call cutting corners or going short roads to places. The short road I have always found is in the end the longest. There are more gates to open, more stiles to get over, something or other to hinder, and the distance we save we lose in the time we take. Set one man to go to a place four miles off by the road; set another to go a short cut across the fields, and ten to one the man on the road gets there first. And it is natural he should, for the road is the legitimate way, the one that has been tried and found the best, and by going straight on it we shall gain time if not distance.

1852: Hunting

When hunting hare with greyhounds, both a good hound and a good rider will never cut a corner. Plenty of examples can be found in the 1850s and 1860s.

The earliest I found is from 1852 in Knightley William Horlock's Letters on the Management of Hounds, where this is the first of two uses on page 208:

About a hundred and fifty horsemen were at once scattered over the downs, riding at the top of their speed, in almost all directions; some following the hounds, but a greater number, not liking the undulating nature of the ground, cutting corners, and hustling each other by cross riding.

Today: Hunting

It is still true today that good riders and good hounds seldom cut corners. The South Canterbury Hunt's Etiquette, Safety & Commonsense instructs riders:

Follow the line that the hounds are taking, never cut corners unless well behind where the hounds are working.

And hounds that cut corners can be penalised in races. The Lower Mainland Whippet Association says of lure coursing:

Put simply, lure coursing is a simulation of the chase. Two or three hounds of the same breed, selected by random draw, line up at the start line and, on the signal from the Huntmaster, are simultaneously "hand-slipped" by the handlers. The dogs follow an erratic course and are judged not only on speed, but on enthusiasm for the lure and the chase, follow (chasing directly behind the lure and not cutting corners), agility (cornering ability, sure-footedness, etc.), and endurance (not running "out of gas" at the end of the course).

A more detailed example of a three-hound race where two cut corners is given in Lure Coursing: What is it and How to Get Ready, Set And Tally-Ho!, and concludes:

Did blue win? Or yellow? Or pink? Certainly if we scored these athletes according to who reached the finish line first, then blue was the winner, but consider this: these are lure coursing sighthounds, canine athletes. There are many aspects in scoring lure coursing, not just who gets there first. These are broken down in five categories--speed, agility, follow, endurance and enthusiasm/overall ability. The judges must not only assess a value for being first to the finish line, but how did the hound get there? Did the hound cut all the corners or run up the middle of the field to intercept the lure or did the hound follow the lure on every curve and straightaway?

Pink was the winner. Why? Look again at our scenario. Blue led out but cut a corner to keep up with pink, yellow cut the corners, and pink passed blue twice, even after blue cut the corner. Blue had more speed at the end therefore was not running at his top speed over the whole course. Yellow did not run the entire course. Pink had the speed to pass blue during the mid-course, on the straight and more than likely would have finished ahead but pink was beginning to make the stop since he was ahead. Blue took the lead because he was 2nd place coming in and had to use the speed burst to grab the "prey<" first. Pink ran the entire course with good follow, speed, agility and endurance. Blue did not use his power and speed till the last push. Yellow was off-course most of the way. Thus, Pink wins.

I presume hounds are not meant to cut corners as they should follow the hare at a "respectful distance" to tire it out rather than catch it. And riders shouldn't cut corners so as to remain far back enough from the dogs so as not to distract them. Cutting corners results in a less than perfect hunt.

A quick search for "cut corners meaning" turns up this answer:

http://www.phrases.org.uk/bulletin_board/48/messages/316.html

"It's a metaphor from driving - not necessarily motor driving, because it also applies to horse-drawn carriages. When you come to a sharp turn in the road, instead of going all the way to the corner and then turning, you can go diagonally across, and "cut the corner off". This saves time, but entails a risk of clipping the kerb and overturning, or being involved in a pile-up with another vehicle. Thus "to cut corners" means to discard normal safe practice in order to get fast results."

I found a pretty reliable answer to this question from a 1997 letter to the Independent answered by the Chief Editor of the OED:

Sir: Ann Hales (Letters, 1 July) asks after the origin of "to cut corners", in your earlier headline "Nemesis of the golden boy who cut too many corners". The OED documents the phrase from 1869, when Mark Twain described a gondolier who "cuts a corner so smoothly, now and then, or misses another gondola by such an imperceptible hair-breadth that I feel myself `scrooching', as the children say, just as one does when a buggy wheel grazes his elbow" (Innocents Abroad, chapter 23). The original use was, therefore, "to pass round a corner as closely as possible"; the modern extended use is recorded from the latter end of the 19th century.

So the original use of the phrase was "to pass round a courner as closely as possible". It's pretty obvious that to do so at a high speed to save time is reckless and dangerous to passengers. From there it's easy to see how the metaphor was extended to encompass any behaviour that may have been done to save time or materials but in the end may not be the safest or wisest route.