Half Inch Squares Graph Paper: Fill & Download for Free

Here are two other methods in addition to those already described.Count the number of line intersections inside your hand outline. The ratio of this to the total number of intersections on the sheet is an estimate of the ratio of the area of your hand outline to the area of the paper.For example, if you have 8.5 x 11 inch paper with four squares to the inch, there are 1,496 intersections (8.5 x 4 x 11 x 4, to be even more precise, you should subtract half of the intersections on the edges of the paper, that is, subtract 77 to get 1,429). Say 500 of them are inside your hand (as other people suggested, you can count faster by making a few rectangles to cover the big areas). The area of your hand outline is close to 500 x 8.5 x 11 / 1,429 (or 1,496) square inches.This is a variant of Monte Carlo integration. In Monte Carlo, you use random points rather than intersection points. But that is to guard against certain special shapes that might preferentially include or exclude intersection points. For your hand, using fixed points is actually more accurate. Technically it is an extreme case of quasi-Monte Carlo; quasi-Monte Carlo adds some randomness around fixed points, you’re just adding zero randomness. Or, if you like you could actually use a computer to generate a bunch of uniformly distributed random points on the paper and count them instead.Obviously counting intersections in similar to counting squares and partial squares, but it’s easier and probably more accurate unless you’re exceptionally careful and good at estimating areas. Plus there is more theory behind it.The second idea is a variant of Buffon’s Needle approach to estimating pi. Put the paper with the outline on the table, and toss a dime or some other small object onto the paper. Toss from far enough away that it can land anywhere on the paper with roughly equal possibility. Record whether it lands completely inside the hand outline, completely outside or on a line. If it doesn’t land on the paper, it doesn’t count. If it lands partly on the paper and partly off, it counts half.Take the ratio of (inside outline toss + tosses on the line / 2) / (tosses completely on the paper + tosses partly on the paper / 2), and multiply that by the area of your paper. With 100 casual tosses this won’t be as accurate as counting intersections, and it makes no use of the graph paper, but it’s clever and more fun.

How large a house? How many custom details? Just a few tweaks to off-the-shelf plans, or is every detail site-specific and involve long discussions with owners?A rule of thumb and then two contrary examples:Rule of thumb: An architect working with clients to develop a vision, a general design, detailed drawings, assisting in the permitting process, specifying the materials, fixtures, appliances, lighting and working with the builder as questions come up and acting as the owner’s agent in dealings with the builder? Figure 15% of the cost of the home. $500,000 house? $75,000 total fees. $2,000,000 house? $300,000 to the architect. YMMV, of course, but that’s a reasonable budget.Two very different examples:I’m an engineer, not an architect, but I can draw lines on graph paper. I drew up my own house plans, very basic, just two plan views (first floor, second floor) and four elevations (one from each side), each on 8–1/2 x 11″ 1/4-inch-square graph paper, like a high school math student would use, and took it down to the building department in my small Alaskan town. They were happy it was actually to scale and not a sketch on a bar napkin (the builder, even more so). And I paid the fee of $360 and that covered everything - the building permit, all the inspections, and the final certificate of occupancy. Unspecified on those plans was the assumption that everything would be built to local and national codes including energy efficiency standards, while, in practice, all the work I did and had contractors do, was better than those minimum standards (higher R-value insulation, tighter air sealing, more efficient applicances, etc). I wasn’t calling out every 5/8″ tie-down J-bolt around the perimeter of the foundation nor the nailing schedule for every sheet or CDX or OSB. I knew where they needed to be there, the builder knew it, the inspector knew it. It all had to be done right, but no one needed it specified on any plans.At the same time, my parents-in-law were building their retirement home. In coastal California. Subject to California’s Title 24 energy-efficiency standards on top of the UBC, height limitations, square footage limits, limits on the number of bedrooms and baths, etc. When I got my permit THE NEXT DAY, for that $360, they were 9 months and $13,000 into the design and permitting process (building permit, coastal permit, well, septic, archelogical survey, etc, etc) and were ONLY HALF DONE. It ended up being a lovely house, with spectacular views (but I have better views because we have glaciated volcanoes and they don’t), carefully selected woods used throughout, thoughtful lighting in each room, all the conveniences and particular layouts they’d always wanted in the kitchen, baths, home office, etc. And they enjoyed the endless back&forth with the architect to consider every little nuance and detail over a 2-year design and permitting process and 1.5 years of construction (which would have driven me crazy). And every single detail - every bolt to secure each seismic brace, which switch would control which light fixture, what quarry to order the limstone tiles with the fossils in it from - was all in the plans, details, schedules and lists that were generated.The day we poured our own foundation, something else happened and I HAD to be done in 9 months. Turns out you can build a 90,000-pound house in 4 days less time than a 9-pound infant.

TL:DR - Because ships are longer than they are broad.Well, as other writers will assuredly have pointed out, some ships did carry their cannons mounted facing forward. Quite a lot of ships in the high Age of Sail, while carrying the bulk of their guns on the broadside, also spared the space and tonnage to mount a few guns facing fore and aft in addition. Or, more correctly, had a couple of guns at each end which would be run out through fore or aft firing ports as well as on the sides.Having just falsified the question, let’s address the underlying premise of it.It does seem, at first glance, as though sailing warships of the late 16th to early 19th Centuries carried all of their cannons on the broadside, facing perpendicular to the direction of travel. Certainly a ship rated as a one-hundred-gunner would have at least forty-eight facing you when you looked at it from the side. This causes a lot of problems for some people; “The guns are all facing the wrong way. You’d have to turn sideways…”Why didn’t those sailing warships just carry their guns facing forwards, so it’s easier to shoot?Lots of reasons. Lots!Firstly - easier? I do think a lot of people believe that. But what would be easier about it? A sailing ship wasn’t like a fighter plane. Rather than being a stable vehicle travelling a straight course, with only seconds to bear on a fast, agile target flashing across it, the sailing warship was a pitching, bucking beast. Firing guns off the bow, meant having to take into account a huge vertical motion; up and down and up and down… constantly. Puts you off your aim a bit, especially with all that rolling side to side as well.Go to the side of the ship and take another go at it. The rolling motion of the ship is actually an easier motion to cope with than the vertical pitching at the ends. Sometimes faster, but the best warships had a relatively slow roll. And because the ship is narrower than it is long the larger angular range of the rolling motion doesn’t necessarily translate into a bigger vertical movement. The limited effect of the pitching on your accuracy when shooting off the side is a bonus too! All of which means that you just deal largely with a vertical motion when ‘firing on the roll’ - almost no yaw - and far less dousing with spray. Lovely!The sailing warship’s enemies moved relatively slowly too. It took far longer to open or close the range (or escape) than it did to put the helm over and turn the ship, open up the broadside and Boom! - have fifty-three cannon balls sir! Then put the helm back over and carry on chasing you.In fact, why are you running away? You, sir, are defeated! If you wanted to fight me, you’d have stayed and fought now, wouldn’t you?So, that ‘fifty-three cannon balls’ thing. Yeah. Imagine, if you will, a ship of two hundred feet long by fifty feet wide (it’s a theoretical ship). Draw it on graph paper; one square to ten feet. Imagine that you can draw one cannon in each box on the ship’s edge. Oh! It’s not fifty feet wide at the ends, either. It suddenly goes all oval-shaped at one end and tapers to a narrow square at the other. You get about three squares width at each end.Okay, so now you should have been able to fit about eighteen guns along each side and three at the ends. Now rub out one of the forward ones, because of the stem-post and bowsprit (things which hold the ship together and help it go). So that’s two, then. And now do the same at the other end, because of the rudder (the thing that makes the ship turn).So you have two at each end and eighteen on each side. Now make the ship a three decker, oh but don;t put any guns on the fore and aft ends of the lower deck, because of the pitching - they’ll get washed out by sea water!What does that make? Four guns shooting forwards, four guns shooting backwards aaand (three times eighteen is…) Fifty-four guns on the broadside. On each broadside. For a hundred and sixteen guns total (like I said, it’s a theoretical ship, not a historical one).And that’s pretty much how a wooden sailing warship was set up, be it a hundred-and-ten, a ninety-eight, a seventy-four gunner or whatever. A couple of bow and stern chasers and the rest on broadside because they fitted there.Honestly, there was no disadvantage to turning side-on to fire at the enemy. If I’m using guns, why do I want to be chasing you down? You’ve come to fight too, right? That means you’re in no great hurry to try and jump aboard my ship; you’ll wait til you’ve smashed me to bits and killed my marines with your guns first! I’ll do the same. If we’re sailing parallel to each other, we can blast each other as long as we care.(Above: In no hurry to punch each other in the face just yet.)When ships are set up like that, armament-wise and formation-wise, it makes sailing straight at me a dangerous and foolish practice. You have two or maybe four guns to shoot at me with, while I have more than ten times the number. And what will you do when you get to me, in your sailing ship? Ram me?! Ha ha, I rather think not!This is why Nelson’s tactic at Trafalgar seemed such a gamble - and it was dangerous!But, ramming, eh? The tactic of the warship which has mechanical propulsion. Like oars, or paddle-wheels, or screw propellers. Yes, the warships with that kind of propulsion… they actually did carry forward facing cannons.(Above: A galley. Yes, I’ve used this picture recently in another answer and I’ll wear it out if I’m not careful.)Galleys, ancient Mediterranean warships which lasted through to the Age of Enlightenment, had sails but relied far more on oars for propulsion. This made them swift, nimble and most of all, powerful. The strength of their oarsmen could put a tremendous amount of energy into the tip of that ram, punching into a target’s hull and wreaking terrible damage. Those same oarsmen could then haul the galley backwards to get it clear of the target, or turn it on the spot to move off swiftly.These were close-range tactics which involved pointing the ship at the enemy. that’s why, when guns were fitted to galleys during the Renaissance, they were mounted forwards. Heavy armaments, too; three, four, or even five of the things, plus as many swivel guns! All of which is a far less powerful armament than a Napoleonic three-decker could muster. Not just in numbers of guns, no. You’ll like this:A three-deck first rate ship of the line in 1800 had a broadside with about eighteen times the throw weight of the best-armed galley’s forward battery. HMS Victory’s lower deck alone had nearly eight times the throw weight of a galley’s five guns (including the massive thirty-pounder on the centreline). No contest.But having forward-facing guns made sense for galleys, which actually would spend their time closing on the enemy, bows-first. What else are you going to do?And when steamers became shell-proof in the 1850s, they also became shot-proof. Slicing head-first into the enemy ship with your own seemed a fair way to damage the foe once again. Bow guns came back into vogue and, coupled with pivots, turntables and turrets, they didn’t have to stay pointing straight ahead either!(Above: Cerbere-class turret rams of the French Navy; late 1860s. Two big, tasty 9.4-inch calibre guns pointing forwards, or in almost any direction you like!)What people found with the late resurgence of rams in the last third of the 19th Century was that these ships would hardly spend any time at all pointing right at their enemy. Either the enemy would be steaming at high speed perpendicular to the ram, which meant the use of ‘deflection ramming’ (steaming ahead of the enemy’s current position, with lots of helm, so as to meet them on a converging course), or the darned enemy would have so much way on that your ram could never even get into position to attack!Most of the time, it was accepted, the steam ram would be a small battleship and would fight using battleship tactics of drawing up parallel to the enemy and maintaining that position, not chasing after the foe like they were Benny Hill and getting left behind! So, the requirement for forward fire in the age of steam never obviated the need for broadside fire. Even the big ironclad battleships started to take this ‘mixed tactic’ approach into their designs, with the all-round fire best expressed by the early 1870s central battery arrangement of artillery:(Above: Danish ironclad Odin, with four guns at the corners of a square central battery. Each gun had two firing ports; one on the broadside and one either ahead or astern. Oh, she also had a retractable ram, for super levels of innovation and cool!)Central batteries made use of the guns in much the same way as turrets did, in providing all-round fire, but of course turrets were a bit more efficient. The irony being that, with turrets and barbettes, a warship didn’t just have better end-on fire than the old sailing ships, with up to half of its guns firing ahead or astern. With barbettes & turrets it had better concentration of weaponry on the broadside too; up to one hundred percent of the main armament!(Above; Baden. Four ahead, four astern, eight on broadside. About as good as it gets.)Even with the very best and heaviest bow fire arrangements, then, it seems that broadside was still just better.