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Three typefaces for mathematics

Video in TIB AV-Portal: Three typefaces for mathematics

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Three typefaces for mathematics
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Part Number
12
Number of Parts
33
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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
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2012
Language
English
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Cork, Ireland

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Abstract
After a brief discussion of some of the typographic and technical requirements of maths composition, three case studies in the development of maths types are presented: Times 4–line Mathematics Series 569, a complement to the Times New Roman text types as set with Monotype equipment; AMS Euler, an experimental design intended to contrast against non–mathematical typefaces set with TeX; and Cambria Math, designed in concert with a new text face to take advantage of new Microsoft solutions for screen display and maths composition. In all three cases, the typefaces were created to show the capabilities of new technological solutions for setting maths. The technical advances inherent in each font are shown to be as central to its function as its visual characteristics. By looking at each typeface and technology in turn, and then comparing and contrasting the issues that are addressed in ! each case, it becomes apparent that even though certain challenges are overcome with technical advances, the need to consider the specific behaviours oftype in a maths setting remains constant.
Mathematics Goodness of fit Presentation of a group Computer animation File format Multiplication sign Configuration space Endliche Modelltheorie Quicksort
Dataflow Context awareness Observational study Multiplication sign Virtual machine Shape (magazine) Mathematics Positional notation Different (Kate Ryan album) Software testing Endliche Modelltheorie Extension (kinesiology) Position operator Physical system Form (programming) Scripting language Series (mathematics) Multiplication Mapping Projective plane Bit Line (geometry) Entire function Numerical analysis Symbol table Type theory Computer animation Personal digital assistant Quicksort Family
Context awareness Multiplication Multiplication sign Source code Sampling (statistics) Time series Chaos (cosmogony) Shape (magazine) Word Mathematics Computer animation Positional notation Different (Kate Ryan album) Mixed reality Quicksort Nichtlineares Gleichungssystem Error message Alpha (investment)
Multiplication sign Projective plane Fitness function Staff (military) Black box Solid geometry Computer font Type theory Mathematics Arithmetic mean Casting (performing arts) Computer animation Average Internet service provider Mixed reality Self-organization Diagram Right angle Extension (kinesiology) Family Row (database) Spacetime
Point (geometry) Slide rule Multiplication sign Shape (magazine) Student's t-test Rule of inference Product (business) Revision control Mathematics Casting (performing arts) Core dump Matrix (mathematics) Damping Diagram Nichtlineares Gleichungssystem Series (mathematics) Contrast (vision) Set theory Task (computing) Physical system Closed set Consistency Bit Line (geometry) System call Type theory Computer animation Order (biology) Cycle (graph theory) Figurate number Row (database)
Context awareness Group action Multiplication sign Shared memory Bit Shape (magazine) Line (geometry) Symbol table Numerical analysis Degree (graph theory) Digital photography Mathematics Computer animation Different (Kate Ryan album) Matrix (mathematics) Pattern language Figurate number Family Set theory Physical system
Point (geometry) Multiplication sign Correspondence (mathematics) Virtual machine Time series Function (mathematics) Field (computer science) Product (business) Frequency Different (Kate Ryan album) Term (mathematics) Authorization Endliche Modelltheorie Nichtlineares Gleichungssystem Exception handling Physical system Collaborationism Standard deviation Mapping Information File format Forcing (mathematics) Bit Line (geometry) American Mathematical Society Instance (computer science) Flow separation Symbol table Type theory Computer animation Personal digital assistant Reading (process) Asynchronous Transfer Mode
Direction (geometry) Characteristic polynomial Maxima and minima Bit Shape (magazine) Connected space Radical (chemistry) Word Process (computing) Computer animation Positional notation Formal grammar Authorization Text editor Quicksort Mathematician Form (programming)
Rotation Area Digital electronics Digitizing Bit Staff (military) Shape (magazine) Mathematical model Skeleton (computer programming) Medical imaging Word Computer animation Angle Right angle Endliche Modelltheorie Personal area network
Pixel Presentation of a group Variety (linguistics) Multiplication sign Coroutine Parameter (computer programming) Graph coloring Neuroinformatik Revision control Medical imaging Mathematics Different (Kate Ryan album) Diagram Software testing Data structure Endliche Modelltheorie Design of experiments Metropolitan area network Scaling (geometry) Texture mapping Mapping Weight Bit Type theory Computer animation Visualization (computer graphics) Phase transition Volumenvisualisierung Metric system Resultant Reading (process) Spacetime Asynchronous Transfer Mode
Point (geometry) Suite (music) Computer file Latin square Multiplication sign 1 (number) Computer font Product (business) Revision control Mathematics Alphabet (computer science) Matrix (mathematics) Endliche Modelltheorie Nichtlineares Gleichungssystem Set theory Metropolitan area network Scripting language Projective plane Bit Symbol table Equivalence relation Computer animation Personal digital assistant Text editor Family Asynchronous Transfer Mode
Slide rule Group action Pixel Touchscreen Diagonal Multiplication sign Projective plane Mathematical analysis Bit Online help Shape (magazine) Computer font Dimensional analysis Type theory Radical (chemistry) Computer animation Artistic rendering Vertex (graph theory) Video game Endliche Modelltheorie Contrast (vision) Family Set theory
Point (geometry) Greatest element Decision theory Combinational logic 1 (number) Open set Shape (magazine) Product (business) Usability Element (mathematics) Revision control Mathematics Different (Kate Ryan album) Natural number Alphabet (computer science) Cuboid Diagram Nichtlineares Gleichungssystem Endliche Modelltheorie Set theory Position operator Dependent and independent variables Mapping Bit Type theory Exterior algebra Film editing Computer animation Text editor Table (information) Active contour model Asynchronous Transfer Mode
Point (geometry) Variety (linguistics) Code Weight Execution unit Mass Mathematics Computer animation Alphabet (computer science) Optics Quicksort Arithmetic progression Position operator
Computer animation Multiplication sign
my presentation is not about technology but it's really about design issues and how they are influenced by technology and how they can sort of throws some issues back to the technology itself and I mean typesetting for a very long time I the that is using a lot of sort of poor configurations for dealing with math as I got further into typeface design really made want tackled the issues will what makes a good typeface format usage I knew that I had identified lots of problems as composite and a designer but I really wanted to look for some models about what the issues involved so was casting about for the things to look at for this and eventually arrived
at this sort of 3 case studies that bring up different kinds of issues on particularly because they're all very tightly related to very specific technologies where they look to these problems for solutions on and the solutions that they arrived at and have become models for other kinds of design really set the stage for what would follow which may be very interesting to look at what actually led to these conventions that happened so I'm when I'm speaking about the times for line mathematics series this is the complete overhaul of the times new roman a typeface family but the Monotype Corporation produced for working with metal type for it's hot metal machine testing system where they developed a new method of setting maps and metal and use that as an opportunity to completely reworked the entire times family and there is always the faster probably more familiar with but but I want to look a little bit more about that in this case the interdependence of the font design and the technologies but because it weighs very experimental typeface ends the the I actually think that some the issues of that experiment could get a little bit more use than they have over the years and in the Cambria family which already talked about as very glad he talked about a lot of technical issues because frees up some time for me to look at the actual shapes of the letters a bit more so the couple a general issues
that came up In all of these projects I think in every attempt to create a typeface that works for mathematics or even to evaluate and choose typefaces for setting that's on this issue of legibility and how a text legibility is a very different animal from the legibility of mathematical notation the problems of combining multiple type styles multiple scripts multiple sizes of letter forms and numerical forms versus symbols all in 1 context in a way that the all clear and then the positioning and spacing issues were is text that we read tends to move in a very steady horizontal flow and mathematical notation away behaves like a script of it sounds it moves vertically and horizontally on ends with To some extent even back and forth a little bit more than you may expect and how we're trained to read normal on the the basic
issue of legibility of text 1st mathematics is that we read text basically by identifying word shapes so when you have an issue of a small typo of more things that don't quite make sense the way your brain interprets all that we just moved past that so in this effort you know have the word mathematics that normally it's set with their and alpha rather than a in the middle and the phrase in parentheses which is sort of move past it in the way we process text those little kinds of errors can completely change the meaning on in mathematical notation of people don't identify the shapes correctly easily of n years probably better than saying well technically at 2 different characters you can tell if you look at them side to side but effect and if those things can be identified more quickly it's probably a good solution for the problem at hand is the sample
equation from the the specimen for the times series when it was redeveloped just showing the mix of multiple kinds of things that happen all 1 context in mathematics and I it's very it's very easy to see them sure if you compose math you know that your pulling together things from different sources all the time and try to fit them together in a way that minimizes the visual chaos of all that stuff happening in 1 place at the same time we're also asking people to switch back and forth sometimes very rapidly between reading text and going into the mathematical notation and trying to work it out so anything that can minimize the issues of that back and forth will help so the time series
and as I said this is a solution for speeding up the composition of math in metal ion and which fork hundreds of years have been done in a very manual way type would be precast and of probably very highly skilled composite would be needed to quickly fit all those pieces together and this is very very expensive and require a lot of organization within type shop and he kept a lot of this specialist work out of the hands of printers you didn't have a skilled staff to deal with that so Monotype is a provider of typesetting equipment I wanted to find a way to give more their customers the ability to take on this kind of work particularly in the wake of World War Two when there was an explosion of technical publications that could really benefit from a better solution to this problem that have a fit the pieces together so any kind of a change to what would happen in providing typefaces for metal typesetting was a huge industrial undertaking it involved the casting equipment the typesetting equipment the creation of the moles that created the middle type to Monotype shows times new roman which was 1 of the best sellers for book work at the time and invested a huge amount of time and money into getting it to work the solution is basically a simple 1 mechanically of but in so much effort had gone preparing the typeface for they took it as an opportunity to rework the typeface for the math usage the and because of the need of every glyph had to be reworked to work with a new means of setting the tide they really had review all of the glyphs that might have been used for MAP which in the past have been pulled from many typefaces that someone they have available in the shop and in the end by the time they really discontinued the family in its phototypesetting age they had redrawn over 8 thousand lives all provided really only a 1 size but it gives you some idea of the extent of the project and this is a
diagram of how mathematical material would be fitted together to be set in middle time every black box is a solid piece of land that needed to be fit together with everything else you could have spaces where there is no material a piece of led 0 it had to be fit in to hold this massive tied together and when I you have material that's not it's not capable of being set in an orderly row you have to start packing things in around it if you're starting to mix sizes of type so you have to pack things around to make sure that your you your 5 . character fits in extra 10 . character all the same learning and and this is the work that you would need some to just do by hand is get all the right pieces identify all the right characters pull them from the font the right size pack them all together or Monotype wanted to work out a method that would automate is much of this is humanly possible of so that of a more than average worker in In shock to take care of it the so they worked out the
solution call before line method where just about I think the figure was about 80 per cent of the material that would show up for math could be set in line so by looking at equation as for rows of material the the core of this solution was to basically set a full-size letter larger than the piece of metal for the call the body of a type slide that it said answered the legally agency X in this diagram those are pieces of 10 point type but instead of being set on a piece of metal that would give you 12 points from baseline to baseline they were set on a 6 point piece of metal with the letter overhanging then that allow them to set things like superiors inferiors of 1st and 2nd order on those half-lines cast on their own so this way instead of pulling together pieces of type that were set from entirely different sets of a tight matrices these could all be loaded up at 1 time set all the same time with a little bit of hand were to put in oversize characters in a horizontal rules that fit in and things like that but by said since they had to create different matrices from the molds that would actually task these pieces of metal type the hair redrawn every 1 of these characters to go through the float production cycle and that given the opportunity to really adapt the typeface I some of the things that
they did were to just improve the consistency of what would get pulled together for men as well as the increased legibility when possible beforehand before the introduction of the 4 lines system the most material at least in the UK recommended the use of Monotype's modern serious 7 and a modern typeface squarish a lot of contrast you can see and the figures in particular shapes that closed up a lot at small sizes and was mixed with available Greeks that didn't necessarily fit the same proportions in seeing in the lines of outlined in red at the top of the x height student quite line originally so when they when they redrew the series 5 6 9 which is this reworked version times new roman is even d'après proportions increase the sizes of what was available for the 2nd order inferiors and superiors to help make them a little bit more legible and also switch to a style of figures used in Times New Roman which in close up quite as much of the small sizes but the
big difference they made a they made throughout the entire set of i italics of the entire family it was a small change but actually made all the difference in fitting these pieces of metal together in a context where you often switching back and forth from and Italic figure to upright number or symbol of these are photographs of the patent drawings that to make the middle matrices would be cut from the and this gives you an idea of
what they did a simple change of changing the slanted that italic 4 degrees making it a little bit more opyright made a huge difference times new roman actually had a pretty dramatic slanted 16 degrees that we got into a lot of characters made it very difficult to combine with other symbols in seeing the original drawing of the basic times new roman that the pattern of an of an F and B were share the acts were murder when they redesigned the family for the for line system this action that 16 different shapes of just the lowercase i talent that they provided to handle all of these situations the and they just kept going and going and
going there were consulting with a lot of technical presses at the time of who course you know they they brought up exceptions all the time and rather than trying to standardize and force forced presses to work in a given way any more than they actually were already doing they tried to remember that these were paying customers and and a lot of cases the customer was right in 1 of the characters that they want to see just the basics that they started out with with the release in 1959 including some notes about the various exceptions that were available on but they still stuck to this idea of trying to get the authors and publishers to work in a given away then the things easier on the system the time series 5 6 9 format still had to be loaded in the hot metal pasta entirely at a separate time then the type that was used to set text which is what this whole notion of highlighted in yellow is all about and they looked largely the same but particularly in italics they were not the same at all and they were definitely set on different sides bodies different sized pieces of let that supported the letters themselves so they had to be set separately and if someone who really and truly wanted an equation in line with text and they had to be cast in metal at different times switching output not machine and fitted together by persons which sort the whole notion really was behind encouraging them to keep them separate it just made it faster and cheaper and easier to produce for the composite A's and and is a pretty successful system when he introduced in the fifties it just swept through the UK and because it opened up the field to so many people and became a standard of how work was approached you know once was adapted for film for about 30 years it is look at books published in the UK around this period there all set times new roman 10 point but there was a similar system that was used in the US and North America of based out of North America and in both of these products were sold all over the world and that was based on the century model the century-type typefaces but the but it's a similar concept of 4 lines getting stacked together these composition and but it still have this restriction of working with metal keeping elements separate and it was in based very much on the concept of the map should look as much like the text as humanly possible so that people are aware of jumping back and forth between these modes of reading these styles of typefaces that were used well the it 2 years later after all this had
become a standard but Donald new had a different idea of about how things should that makes the how they could get minutes and with the tools that he had written to put into play the creation of typefaces composition but it was very very easy to pull together different typefaces of symbols drawn from different forces and switching between modes within a text aside from just the capabilities that he'd written tech to do this this is a style of reading and presenting information that he was interested in that he favored and experimented with them and this is this is definitely a movement away from traditional book topography and when I when Hermann Zapf was approached to develop typefaces I he knew very much had a kind a collaboration on the concept of how this to be approached Hermann Zapf all did the drawings that the typefaces were based on eventually but there's a very very it on passionate correspondence between Zappa news about what this new typeface could be and they didn't feel hindered by the idea of making sure that had to match what it was going to be used with they could try different model a a new approach and what they did in the course of this has come up with a very innovative design when compared with traditional book topography and instance this is being designed at the same time that the actual technology was being defined in but and the design that terms that came up with that of the American Mathematical Society news really were very fond of and they took that as a reason to help push the technology further a little bit to really express the designed well and make sure that it can be presented the way was conceived the I so the basic idea that arose so
after the initial discussions was upper-right I talent shape of letter but a casual form that reflected the tone of hand writing that mathematicians use and and working with the traditional notion of the letters being used in mathematical material being set my talent but just eliminating the problem of how to actually fit all the pieces together with a slanted shape and they turned it up right there and then it's hiding its its hung on to the characteristics of an italic typeface With that sort of minimal use Seretan certain shapes to the terminals in a bit of a casual shape to the letter of and he took another step further with this notion of a calligraphic idea but a very casual kind calligraphy 1 of the really interesting notes that was brought back to zap which he really got behind was this notion that you don't want to use without trying to capture the sense of fine formal of broadening pen calligraphy want captures some of the immediate casual feel of some very quickly writing out the notation to express their ideas says after work with that refined it and we're left with this design which is very unlike the kind by italics that would have been used in books previously even though both he induced wanted to start with ideas from book topography really really analyze the problem they moved away from it in the design of which really opened up this experiment in makin looking a very different way than it had been before of and it's just
an example I have of you even add and editorial manuscript word then they do the authors in the editors direct connection to the material very often was through hand writing all of this simple typesetting techniques I didn't give you access to all of these so there was a kind of familiarity already with seemed shapes like this that they wanted to rather than buried in the process make it a little bit more visible to the user the the trick of course was
taking the subtlety of a typeface design like the expressed like the 1 expressed and that drawing and rendering properly in fairly primitive digital technology primitive is a little bit too misleading a word technology that was still being defined still being improved 1 of the 1st things that they realized that when the team at Stanford try digitizing zap drawings was the basic conceptual model of Metafont wrappings stroke styles around basic skeleton didn't really capture everything that was in zaps design so we go back the a little bit of
these drawings that the image on the right that you have if you haven't seen any of these of these before are there the full-size drawings that staff delivers for digitization there's a little bit more going on in those outlined shape than the basic skeleton with mathematical model of the Pan applied on his opening up some of the areas where strokes come together to keeping from building up there are some subtle modulation with the things I they are conceptual idea of the pain creating that shape as a little bit more rotation and what's going on with the pen and just setting an angle and expressing along the still so
1 of the 1st things that they realize that they were going to have to do was rather than working off the center of the strokes I work with Metafont in a slightly different way In creepy outlines in concreteness create a structure that wrapped around the edge of the character could be manipulated that way I after that phase is done they still had to get routines that could generate all the bit maps off of those outlines of the time that this is all done everything was being renders bit maps in the version that that would go press wasn't being done quite on the fly in an image set in the same way that PostScript technology would do later on I'm so there's a lot of time that went into a fine tuning the proportions of these characters and spacing in analyzing what what happens when these were rendered at different scales just to get evenness across the stem weights of all of the strokes that 1 pixel difference that's been pointed out this diagram from David Siegel's book really changes the overall look of text when you see it full size and In a anywhere between 5 and 10 . those just read as different unlike shapes because of that single pixel and the same the same notion of 1 . shifting 1 side or another also affected the spacing was the spacing test which of them trying to get the metrics right to produce an even texture and they were the team at Stanford I was doing this I was trying to work at general principles so that a user could still specified this type being used in a variety of sizes and get it to work so they were trying to really work at the parameters of what would happen so there's this interesting back and forth In this typeface design not only as a visual design experiment new approach the typefaces but also an experiment of how to make this technology to grapple with the with typeface designs that had a little bit more subtlety to them and ask more of the technology and is a really nice result of it came out with you can see this is for an example
from the 1st news of of on the typeface on concrete mathematics where news worked out a slightly different version of of Computer Modern to work with the overall color but it's still working with this idea if you're really visually switching modes from text to man as a very straightforward clue that through the material that you're reading is changing from 1 type of thing to another and it demanded a different style of reading I'm and there were the actually I'm very fond of this design and I wish this concept had gotten a little bit more use over the years I find looking at material I see things like this show up once a lot and I find it very refreshing I but it's still not very traditional ends a lot of other typefaces that can used for math that that is designed for men back away from this model and go a little bit back towards times model of having things that are very tightly related and following a more traditional dumb presentation of the characters that can used inside the so found the
is just an example of the basic set of characters that was actually designed and packaged in the or the files themselves the effectively very limited just a few alphabets that were designed to work together the basic Latin and Greek characters that can have immediate equivalence in the Latin character set were know also the ones the 1 you use they didn't bother with on Fraktur alphabet and a script alphabet that were designed a little bit more simply than their traditional models in a way that worked better with the other alphabets that were created all other technical symbols are assumed to be pulled from other fonts which technique so simple which in itself was an incredible change from previous ways of typesetting mathematics where everything really had to be fitted altogether on whether it be on a film strips or matrix case so jumping ahead yet again
we have of Cambrian man which is the showcase for mathematical typesetting from on from Microsoft ClearType project again I'm very glad that all cover a lot of the technical issues going on behind this so we can look at some of the other things that came up the Cambria this is very very clearly intended to be a replacement for times new roman suite of Microsoft products into that and the basic concept driving it was not to rock the boat too much on the and if you look at the 2 typefaces next to 1 another they're very very different but Cambridge is like Times Roman has really become over the years particularly in its In its digital versions of very very evenly textured fairly neutral typeface and 1 of the big things the Microsoft wanted to do when they made the switch was they didn't wanna have anything to experimental but you know this is for consumer products everyday users they wanted to keep it pretty straightforward not could give people any big surprises so when they chose Cambridge to be showcase of the mathematical tools but they're going to make available they wanted to make sure that air everything worked together very very smoothly they designed a huge family that would match the text and because they didn't want people so we wondering what would happen when they switched into math mode within Microsoft products a lovely idea which is immediately thrown out if someone is working in a different typeface and suddenly in certain equations and using the equation editor where will just at this point to fall to Cambridge until more typefaces available to take advantage of the tools the the
basic driving concept for the look of Cambridge were the possibilities and the restrictions of clear type rendering which is a new technology for showing how typefaces would look on screen this on the fly transition from the underlying outline of the typeface design to have a pixels screen would show it and what clear type basically made possible was to use every 1 of the sub pixels of an screen letter to show the shape of that letter conceptually should single if but more specific rather than just thinking of is this black or white is on or off I'm going across the horizontal dimensions of the screen it said we actually can switch on or off the red green or blue channels on and get a little bit or life a little bit more definition in this 1 access access was only and yelling balls who was that the chief designer the camera family is probably 1 of a small set of people in the world from really really gets on-screen hinting his work with a lot of he did a lot of analysis at the start of this project for what kind of shapes really respond well to this rendering model and Cambria is his his answer to that problem said about how do we not only come up with a new typeface designs relatively neutral but 1 that really shows off this new technology and it's possible so he arrived at shapes that are somewhat rectangular in overall feeling and as he describes it curves that move very quickly from the horizontal to the vertical on squared off squared off terminals respond very well there is very very sharp increase but the rendering model and they also very very even stem widths and regular spacing help a lot because they'll hit they're more likely to hit like channels In the rendering as it moves across horizontally and in general a fair amount of contrast between horizontal strokes and vertical strokes but also come up crisper and much easier to read at small sizes so you avoided by diet large diagonal gestures wherever possible in even if you have something like an italic character it's pretty squared off of the the the so a lot of
the I think this slide probably is where let's just saved a lot of times and I have to explain quite as much what's going on with it but OK are I think can jump through so all of this stuff working into the typeface features but hyphae still took advantage of all
these alternatives that were available with open type on there's a text mode and a map node except the all wrapped up in the same typeface now and it's relying on the technology that setting the math to decide which is necessary the 2nd you engage the equation editor within Microsoft products it's going to not take the glyphs set of text set and pull the ones from the maths and you can see how the alternates work of the math alternates were the ones designed by Ross Mills up to have a few more visual cues that this is a very very different letter from the upright version or from other alphabets used to in equations you can also see in the Greek on the the top is the text model the bottom is the MAP version where was drawn to be a little bit more upright ease with the combination of other elements and this just
a little bit of a diagram of the of how box model this impl implemented in in type where it actually puts a lot of responsibility for getting these features right and gives it to the typeface designer instead of a composite at this point is as the typeface designer to look at the shapes that he's putting together and make some decisions upfront about how the typesetting technology will fit the pieces together the OpenType tables actually allow you to define what the nature hours of all these and what can be quite complex cuttings around the outer contours of these letters and as the 2 as the typesetting tool position something vertically it analyzes where the cuttings go to figure out how closely the pieces can fit together which allows it to work regardless of the shape that's been drawn for the typeface on and I have no doubt that different typeface designs excellent mention of going through a very different issues for how these hooks within the OpenType table work but I'm really eager see other typefaces they use it to see how the solutions are arrived at of a
large variety of optical sizes were built into Cambrian mapped to provide different starting points for any kind scaling that will be needed in the typefaces and and it's not just a linear per progression can see that this sort of are some the features increase a decrease in size and weight it's deemed appropriate for where the meant to say in in the vertical positioning the
of lots and lots of characters or mentioned in all of these alphabets shown at the bottom every 1 of these glyphs is a separately encoded unit code character there is a separate code for a sans-serif mass go left and they're all written that way into camera math so these aren't stylistic changes these are the separately encoded glyphs within the typeface that the math engine will actually pull in as needed and thank you the and
started to print so much a little time but I hope in some sense without being
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