## My All 2020 Mathematics A to Z: Extraneous Solutions

Iva Sallay, the kind author of the Find the Factors recreational mathematics puzzle, suggested this topic for the letter X. It’s a fun chance to look at some of the basics of (high school) algebra again.

# Extraneous Solutions.

When developing general relativity, Albert Einstein created a convention. He’s not unique in that. All mathematicians create conventions. They use shorthand for an idea that’s complicated or common. Relatively unique is that other people adopted his convention, because it expressed an idea compactly. This was in working with tensors, which look somewhat like matrixes and have a lot of indexes. In the equations of general relativity you need to take sums over many combinations of values of these indexes. What indexes there are are the same in most every problem. The possible values of the indexes is constant, problem to problem, too.

So Einstein saved himself writing, and his publishers from typesetting, a lot of redundant writing. This by writing out the conditions which implied “take the sums over these indexes on this range”. This is good for people doing general relativity, and certain kinds of geometry. It’s a problem only when an expression escapes its context. When it’s shown to a student or someone who doesn’t know this is a differential-geometry problem. Then the problem becomes confusing, and they can’t work on it.

This is not to fault the Einstein Summation Convention. It puts common necessary scaffolding out of the way and highlighting the interesting unique parts of a problem. Most conventions aim for that. We have the hazard, though, that we may not notice something breaking the convention.

And this is how we create extraneous solutions. And, as a bonus, to have missing solutions. We encounter them with the start of (high school) algebra, when we get used to manipulating equations. When we solve an equation what we always want is something clear, like

$x = 2$

But it never starts that way. It always starts with something like

$x^3 - 8x^2 + 24x - 32 + 22\frac{1}{x} = \frac{6}{x}$

or worse. We learn how to handle this. We know that we can do six things that do not alter the truth of an equation. We can regroup terms in the equation. We can add the same number to both sides of the equation. We can multiply both sides of the equation by some number besides zero. We can add zero to one side of the equation. We can multiply one side of the equation by 1. We can replace one quantity with another that has the same value. That doesn’t sound like a lot. It covers more than it seems. Multiplying by 1, for example, is the same as multiplying by $\frac{x}{x}$. If x isn’t zero, then we can multiply both sides of the equation by that x. And x can’t be zero, or else $\frac{x}{x}$ would not be 1.

So with my example there, start off by multiplying the right side by 1, in the guise $\frac{x}{x}$. Then multiply both sides by that same non-zero x. At this point the right-hand side simplifies to being 6. Add a -6 to both sides. And then with a lot of shuffling around you work out that the equation is the same as

$(x - 2)^4 = 0$

And that can only be true when x equals 2.

It should be easy to catch spurious solutions creeping in. They must result from breaking a rule. The obvious problem is multiplying — or dividing — by zero. We expect those to be trouble. Wikipedia has a fine example:

$\frac{1}{x - 2} = \frac{3}{x + 2} - \frac{6x}{(x - 2)(x + 2)}$

The obvious step is to multiply this whole mess by $(x - 2)(x + 2)$, which turns our work into a linear equation. Very soon we find the solution must be $x = -2$. Which would make at least two of the denominators in the original equation zero. We know not to want that.

The problems can be subtler, though. Consider:

$x - 12 = \sqrt{x}$

That’s not hard to solve. Multiply both sides by $x - 12$. Although, before working out $\sqrt{x}\cdot(x - 12)$ substitute that $x - 12$ with something equal to it. We know one thing is equal to it, $\sqrt{x}$. Then we have

$(x - 12)^2 = x$

It’s a quadratic equation. A little bit of work shows the roots are 9 and 16. One of those answers is correct and the other spurious. At no point did we divide anything, by zero or anything else.

So what is happening and what is the necessary rhetorical link to the Einstein Summation Convention?

There are many ways to look at equations. One that’s common is to look at them as functions. This is so common that we’ll elide between an equation and a function representation. This confuses the prealgebra student who wants to know why sometimes we look at

$x^2 - 25x + 144 = 0$

and sometimes we look at

$f(x) = x^2 - 25x + 144$

and sometimes at

$f(x) = x^2 - 25x + 144 = 0$

The advantage of looking at the function which shadows any equation is we have different tools for studying functions. Sometimes that makes solving the equation easier. In this form, we’re looking for what in the domain matches with something particular in the range.

And now we’ve reached the convention. When we write down something lke $x^2 - 25x + 144$ we’re implicitly defining a function. A function has three pieces. It has a set called the domain, from which we draw the independent variable. It has a set called the range. It has a rule matching elements in the domain to an element in the range. We’ve only given the rule. What are the domain and what’s the range for $f(x) = x^2 - 25x + 144$?

And here are the conventions. If we haven’t said otherwise, the domain and range are usually either the real numbers or the complex numbers. If we used x or y or t as the independent variable, we mean the real numbers. If we used z as the independent variable, and haven’t already put x and y in, we mean the complex numbers. Sometimes we call in s or w or another letter; never mind that. The range can be the whole set of real or complex numbers. It does us no harm to have too large a range.

The domain, though. We do insist that everything in the domain match to something in the range. And, like, $\frac{1}{x - 2}$? That can’t mean anything if x equals 2.

So we take an implicit definition of the domain: it’s all the real numbers for which the function’s rule is meaningful. So, $\frac{1}{x - 2}$ would have a domain “real numbers other than 2”. $\frac{6x}{(x - 2)(x + 2)}$ would have a domain “real numbers other than 2 and -2”.

We create extraneous solutions — or we lose some — when our convention changes the domain. An extraneous solution is one that existed outside the original problem’s domain. A missing solution is one that existed in an excised part of the domain. To go from $x^2 = 4x$ to $x = 4$ by dividing out x is to cut $x = 0$ out of the space of possible solutions.

A complaint you might raise. What is the domain for $x - 12 = \sqrt{x}$? Rewrite that as a function. $f(x) = x - 12 - \sqrt{x}$ would seem to have a domain “x greater than or equal to 0”. The extraneous solution is $x = 9$, a number which rumor has it is greater than or equal to 0. What happened?

We have to take that equation-handling more slowly. We had started out with

$x - 12 = \sqrt{x}$

The domain has to be “x is greater than or equal to 0” here. All right. The next step was multiplying both sides by the same quantity, $x - 12$. So:

$(x - 12)(x - 12) = \sqrt{x}(x - 12)$

The domain is still “x is greater than or equal to 0”. The next step, though, was a substitution. I wanted to replace the $(x - 12)$ on the right with $\sqrt{x}$. We know, from the original equation, that those are equal. At least, they’re equal wherever the original equation $x - 12 = \sqrt{x}$ is true. What happens when $x = 9$, though?

$9 - 12 = \sqrt{9}$

We start to see the catch. 9 – 12 is -3. And while it’s true that -3 squared will be 9, it’s false that -3 is the square root of 9. The equation $x - 12 = \sqrt{x}$ can only be true, for real numbers, if $\sqrt{x}$ is nonnegative. We can make this rigorous with two supplementary functions. Let me call $g(x) = x - 12$ and $h(x) = \sqrt{x}$.

$h(x)$ has an implicit domain of “x greater than or equal to 0”. What’s the domain of $g(x)$? If $g(x) = h(x)$, like we said it does, then they have to agree for every x in either’s domain. So $g(x)$ can’t have in its domain any x for which $h(x)$ isn’t defined. So the domain of $g(x)$ has to be “x for which x – 12 is greater than or equal to 0”. And that’s “x greater than or equal to 12”.

So the domain for the original equation is “x greater than or equal to 12”. When we keep that domain in mind, the extraneous nature of $x = 9$ is clear, and we avoid trouble.

Not all extraneous solutions come from algebraic manipulations. Sometimes there are constraints on the problem, rather than the numbers, that make a solution absurd. There is a betting strategy called the martingale. This amounts to doubling the bet every time one loses. This makes the first win balance out all the losses leading to it. This solution fails because the player has a finite wallet, and after a few losses any player hasn’t got the money to continue.

Or consider a case that may be legend. It concerns the Apollo Guidance Computer. It was designed to take the Lunar Module to a spot at zero altitude above the moon’s surface, with zero velocity. The story is that in early test runs, the computer would not avoid trajectories that dropped to a negative altitude along the way to the surface. One imagines the scene after the first Apollo subway trip. (I have not found a date when such a test run was done, or corrections to the code ordered. If someone knows, I’d appreciate learning specifics.)

The convention, that we trust the domain is “everything which makes sense”, is not to blame here. It’s normally a good convention. Explicitly noting the domain at every step is tedious and, most of the time, unenlightening. It belongs in the background. We also must check our possible solutions, and that they represent things that make sense. We can try to concentrate our thinking on the obvious interesting parts, but must spend some time on the rest also.

I am surprised to be so near the end of the 2020 A-to-Z, and to 2020, I hope. This and all the other glossary essays for the year should be at this link. All the essays from every A-to-Z series should be at this link. Thank you for reading.

## My All 2020 Mathematics A to Z: Delta

I have Dina Yagodich to thank for my inspiration this week. As will happen with these topics about something fundamental, this proved to be a hard topic to think about. I don’t know of any creative or professional projects Yagodich would like me to mention. I’ll pass them on if I learn of any.

# Delta.

In May 1962 Mercury astronaut Deke Slayton did not orbit the Earth. He had been grounded for (of course) a rare medical condition. Before his grounding he had selected his flight’s callsign and capsule name: Delta 7. His backup, Wally Schirra, who did not fly in Slayton’s place, named his capsule the Sigma 7. Schirra chose sigma for its mathematical and scientific meaning, representing the sum of (in principle) many parts. Slayton said he chose Delta only because he would have been the fourth American into space and Δ is the fourth letter of the Greek alphabet. I believe it, but do notice how D is so prominent a letter in Slayton’s name. And S, Σ, prominent in both Slayton and Schirra’s.

Δ is also a prominent mathematics and engineering symbol. It has several meanings, with several of the most useful ones escaping mathematics and becoming vaguely known things. They blur together, as ideas that are useful and related and not identical will do.

If “Δ” evokes anything mathematical to a person it is “change”. This probably owes to space in the popular imagination. Astronauts talking about the delta-vee needed to return to Earth is some of the most accessible technical talk of Apollo 13, to pick one movie. After that it’s easy to think of pumping the car’s breaks as shedding some delta-vee. It secondarily owes to school, high school algebra classes testing people on their ability to tell how steep a line is. This gets described as the change-in-y over the change-in-x, or the delta-y over delta-x.

Δ prepended to a variable like x or y or v we read as “the change in”. It fits the astronaut and the algebra uses well. The letter Δ by itself means as much as the words “the change in” do. It describes what we’re thinking about, but waits for a noun to complete. We say “the” rather than “a”, I’ve noticed. The change in velocity needed to reach Earth may be one thing. But “the” change in x and y coordinates to find the slope of a line? We can use infinitely many possible changes and get a good result. We must say “the” because we consider one at a time.

Used like this Δ acts like an operator. It means something like “a difference between two values of the variable ” and lets us fill in the blank. How to pick those two values? Sometimes there’s a compelling choice. We often want to study data sampled at some schedule. The Δ then is between one sample’s value and the next. Or between the last sample value and the current one. Which is correct? Ask someone who specializes in difference equations. These are the usually numeric approximations to differential equations. They turn up often in signal processing or in understanding the flows of fluids or the interactions of particles. We like those because computers can solve them.

Δ, as this operator, can even be applied to itself. You read ΔΔ x as “the change in the change in x”. The prose is stilted, but we can understand it. It’s how the change in x has itself changed. We can imagine being interested in this Δ2 x. We can see this as a numerical approximation to the second derivative of x, and this gets us back to differential equations. There are similar results for ΔΔΔ x even if we don’t wish to read it all out.

In principle, Δ x can be any number. In practice, at least for an independent variable, it’s a small number, usually real. Often we’re lured into thinking of it as positive, because a phrase like “x + Δ x” looks like we’re making a number a little bigger than x. When you’re a mathematician or a quality-control tester you remember to consider “what if Δ x is negative”. From testing that learn you wrote your computer code wrong. We’re less likely to assume this positive-ness for the dependent variable. By the time we do enough mathematics to have opinions we’ve seen too many decreasing functions to overlook that Δ y might be negative.

Notice that in that last paragraph I faithfully wrote Δ x and Δ y. Never Δ bare, unless I forgot and cannot find it in copy-editing. I’ve said that Δ means “the change in”; to write it without some variable is like writing √ by itself. We can understand wishing to talk about “the square root of”, as a concept. Still it means something else than √ x does.

We do write Δ by itself. Even professionals do. Written like this we don’t mean “the change in [ something ]”. We instead mean “a number”. In this role the symbol means the same thing as x or y or t might, a way to refer to a number whose value we might not know. We might not care about. The implication is that it’s small, at least if it’s something to add to the independent variable. We use it when we ponder how things would be different if there were a small change in something.

Small but not tiny. Here we step into mathematics as a language, which can be as quirky and ambiguous as English. Because sometimes we use the lower-case δ. And this also means “a small number”. It connotes a smaller number than Δ. Is 0.01 a suitable value for Δ? Or for δ? Maybe. My inclination would be to think of that as Δ, reserving δ for “a small number of value we don’t care to specify”. This may be my quirk. Others might see it different.

We will use this lowercase δ as an operator too, thinking of things like “x + δ x”. As you’d guess, δ x connotes a small change in x. Smaller than would earn the title Δ x. There is no declaring how much smaller. It’s contextual. As with δ bare, my tendency is to think that Δ x might be a specific number but that δ x is “a perturbation”, the general idea of a small number. We can understand many interesting problems as a small change from something we already understand. That small change often earns such a δ operator.

There are smaller changes than δ x. There are infinitesimal differences. This is our attempt to make sense of “a number as close to zero as you can get without being zero”. We forego the Greek letters for this and revert to Roman letters: dx and dy and dt and the other marks of differential calculus. These are difficult numbers to discuss. It took more than a century of mathematicians’ work to find a way our experience with Δ x could inform us about dx. (We do not use ‘d’ alone to mean an even smaller change than δ. Sometimes we will in analysis write d with a space beside it, waiting for a variable to have its differential taken. I feel unsettled when I see it.)

Much of the completion of work we can credit to Augustin Cauchy, who’s credited with about 800 publications. It’s an intimidating record, even before considering its importance. Cauchy is, per Florian Cajori’s History Mathematical Notations, one of the persons we can credit with the use of Δ as symbol for “the change in”. (Section 610.) He’s not the only one. Leonhardt Euler and Johann Bernoulli (section 640) used Δ to represent a finite difference, the difference between two values.

I’m not aware of an explicit statement why Δ got the pick, as opposed to other letters. It’s hard to imagine a reason besides “difference starts with d”. That an etymology seems obvious does not make it so. It does seem to have a more compelling explanation than the use of “m” for the slope of a line, or $\frac{\Delta y}{\Delta x}$, though.

Slayton’s Mercury flight, performed by Scott Carpenter, did not involve any appreciable changes in orbit, a Δ v. No crewed spacecraft would until Gemini III. The Mercury flight did involve tests in orienting the spacecraft, in Δ θ and Δ φ on the angles of the spacecraft’s direction. These might have been in Slayton’s mind. He eventually flew into space on the Apollo-Soyuz Test Project, when an accident during landing exposed the crew to toxic gases. The investigation discovered a lesion on Slayton’s lung. A tiny thing, ultimately benign, which discovered earlier could have kicked him off the mission and altered his life so.

Thank you all for reading. I’m gathering all my 2020 A-to-Z essays at this link, and have all my A-to-Z essays of any kind at this link. Here is hoping there’s a good week ahead.

## Reading the Comics, April 10, 2015: Getting Into The Story Problem Edition

I know it’s been like forever, or four days, since the last time I had a half-dozen or so mathematically themed comic strips to write about, but if Comic Strip Master Command is going to order cartoonists to give me stuff to write about I’m not going to turn them away. Several seemed to me about the struggle to get someone to buy into a story — the thing being asked after in a word problem, perhaps, or about the ways mathematics is worth knowing, or just how the mathematics in a joke’s setup are presented — and how skepticism about these things can turn up. So I’ll declare that the theme of this collection.

Steve Sicula’s Home And Away started a sequence on April 7th about “is math really important?”, with the father trying to argue that it’s so very useful. I’m not sure anyone’s ever really been convinced by the argument that “this is useful, therefore it’s important, therefore it’s interesting”. Lots of things are useful or important while staying fantastically dull to all but a select few souls. I would like to think a better argument for learning mathematics is that it’s beautiful, and astounding, and it allows you to discover new ways of studying the world; it can offer all the joy of any art, even as it has a practical side. Anyway, the sequence goes on for several days, and while I can’t say the arguments get very convincing on any side, they do allow for a little play with the fourth wall that I usually find amusing in comics which don’t do that much.

## Reblog: The Math That Saved Apollo 13

GCDXY here presents images from the Apollo 13 flight checklist. This is itself a re-representing of images that Gizmodo posted when Apollo 13 Commander James Lovell sold the checklist last year, but I’m just coming across this now. And it nicely combines the mathematics and the space history interests I so enjoy.

The particular calculations done here were shown in one of many, many, outstanding scenes in the movie Apollo 13. However, the movie presents the calculations as being done on slide rule, when the computations needed are mostly addition and subtraction. It is possible to use slide rules to do addition and subtraction, but that’s really the hard way to do it; slide rules are for multiplication, division, and raising numbers to powers.

But considerable calculation for Apollo (and Gemini, and Mercury) was done without electronic computers, and the movie would have missed out on presenting an important point if it didn’t have the scene. So the movie achieved that strange state of conveying something true about what happened by showing it in a way it all but certainly did not.

Two hours after a service module’s oxygen tank explosion on Apollo 13, Commander James Lovell did calculations that helped put the ship back on course so that they could return back to Earth. They needed to establish the right course to use the Moon’s gravity to get back home. Check out the article on Gizmodo from November 2011.

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## Wednesday, June 6, 1962 – Food Contract, Boilerplate Purchase

The Manned Spacecraft Center has awarded to the Whirlpool Corporation Research Laboratories of Saint Joseph, Michigan, a contract to provide the food and waste management systems for Project Gemini. Whirlpool is to provide the water dispenser, food storage, and waste storage devices. The food and the zero-gravity feeding devices, however, are to be provided by the United States Army Quartermaster Corps Food and Container Institute, of Chicago. The Life Systems Division of the Manned Spacecraft Center is responsible for directing the program.
Continue reading “Wednesday, June 6, 1962 – Food Contract, Boilerplate Purchase”

## Friday, June 1, 1962 – Operations Coordination Meeting, Astronaut Applications Close

Today’s was the first spacecraft operations coordination meeting. Presented at it was a list of all the aerospace ground equipment required for Gemini spacecraft handling and checkout before flight.

June 1 was also the nominal closing date for applications to be a new astronaut. Applications were opened April 18. The plan is to select between five and ten new astronauts to augment the Mercury 7.

McDonnell has subcontracted the parachute landing system to be used for the first Gemini flight to Northrop Ventura. The cost is estimated at $1,829,272. The design for this flight is to use a single parachute system, a ring-sail parachute with diameter 84.2 feet. Later flights are to use the paraglider system under development. Earlier meetings have worked out a provisional schedule of events for the parachute landing. ## Tuesday, May 15, 1962 – Ejection seat in review; rocket catapult contract; new liaison The first ejection seat design review has been completed. The two-day conference at McDonnell in Saint Louis was attended by representatives of McDonnell, Northrop Ventura (formerly Radioplane), Weber Aircraft, and the Manned Spacecraft Center. This is the first of a series of ejection seat design meetings planned from March 29. Continue reading “Tuesday, May 15, 1962 – Ejection seat in review; rocket catapult contract; new liaison” ## Friday, May 4, 1962 – Threats to First Gemini Spacecraft Schedules Identified The Manned Spacecraft Center has issued its third analysis of the schedule of the Gemini program. The new plan calls for two additional boilerplate spacecraft, in order to help ground testing. Test hardware has begun construction, and plans for the spacecraft ground tests are formed. Continue reading “Friday, May 4, 1962 – Threats to First Gemini Spacecraft Schedules Identified” ## Tuesday, May 1, 1962 – Digital coding, Gemini rendezvous and radar schemes A working group made of representatives from Goddard Space Flight Center and the Manned Spacecraft Center has formed to study making Project Gemini telemetry be transmitted fully by Pulse Code Modulation (PCM) systems. This follows a briefing from Lockheed on the system. Pulse Code Modulation is a method of numerical representations of samplings of an original analog signal. Human speech has been transmitted on such systems in experimental equipment as long ago as 1926, and was used — in conjunction with a vocoder, developed by Bell Labs, and with randomized thermal noise recorded by the Muzak Corporation — to provide secure high-level Allied communications during the Second World War. Continue reading “Tuesday, May 1, 1962 – Digital coding, Gemini rendezvous and radar schemes” ## Friday, April 27, 1962 – Paraglider Meeting Produces 21 Changes Following a review of the design and testing philosophy for the Half-Scale Test Vehicle, part of Phase II-A of the Paraglider Development Program, the Half Scale Test Vehicle Design Review Board has recommended to North American 21 changes in the test vehicle design and the test procedures. Continue reading “Friday, April 27, 1962 – Paraglider Meeting Produces 21 Changes” ## Thursday, April 19, 1962 – IBM Awarded Computer Contract McDonnell has awarded IBM’s Space Guidance Center, of Owego, New York, a$26.6 million subcontract for the Gemini spacecraft computer system. IBM is also responsible for integrating this digital computer with the spacecraft’s systems and the components electrically connected to it. These components are to include the inertial platform, the rendezvous radar, the time reference system, the digital command system, the data acquisition system, the electronics for attitude control and maneuvers, the autopilot for the launch vehicle, console controls, displays, and aerospace ground equipment.
Continue reading “Thursday, April 19, 1962 – IBM Awarded Computer Contract”

## Wednesday, April 18, 1962 – Astronaut Applications Open

NASA is accepting applications for additional astronauts and will be doing so through June 1, 1962. The plan is to select between five and ten new astronauts to augment the existing corps of seven. The new astronauts will support Project Mercury operations, and go on to join the Mercury astronauts in piloting the Gemini spacecraft.
Continue reading “Wednesday, April 18, 1962 – Astronaut Applications Open”

## Friday, April 13, 1962

A report is being presented today to the Gemini Project Office regarding the abort criteria for the malfunction detection system. The report is presented by Martin-Baltimore and the Air Force Space Systems Division.

## Thursday, April 12, 1962

The Manned Spacecraft Center has confirmed that for the currently planned missions the Agena target satellite’s planned orbital lifetime of five days will be sufficient.

## Monday, April 9, 1962

The Gemini Project Office has received the “Gemini Manufacturing Plan” prepared by McDonnell. The plan calls for the construction of four static articles to be used in ground testing. According to the plan, dated April 6, and presented by Earl Whitlock of McDonnell, production spacecraft Number 1 is to be followed by static article Number 1.
Continue reading “Monday, April 9, 1962”

McDonnell has awarded a $1 million subcontract to the ACF Electronics Division of ACF Industries (Riverdale, California). The subcontract is to provide C-band and S-band radar beacons for the Gemini craft. The beacons are to be part of the tracking system for the spacecraft. The C-band radar would transmit at 5765 MHz and receive at 5690. The S-band radar would transmit at 2910 MHz and receive at 2840. ## Wednesday, April 4, 1962 The Defense Products Division of B F Goodrich Company of Akron, Ohio, has been awarded a cost-plus-fixed-free contract by the Manned Spacecraft Center. The contract, for$209,701, is to design, develop, and fabricate prototype pressure suits. Goodrich has been at work on contract-related materials since the 10th of January.
Continue reading “Wednesday, April 4, 1962”

## Tuesday, April 3, 1962

NASA’s Ames Research Center looks to have a place in the development of Project Gemini. Representatives of Ames, and the Manned Spacecraft Center, and Martin, and McDonnell meet today to discuss the Gemini wind tunnel program and the role Ames will have in it.
Continue reading “Tuesday, April 3, 1962”

## Saturday, March 31, 1962

The configuration of the Gemini spacecraft is formally frozen. McDonnell has been defining the spacecraft since the basic configuration was firmed up on December 22, 1961. Since then McDonnell has been writing detailed specifications for the entire vehicle, its major subsystems, and its performance.

## Friday, March 30, 1962

Martin-Baltimore has submitted to the Air Force Space Systems Division the document Description of the Launch Vehicle for the Gemini Spacecraft. This defines the concept and the philosophy for each proposed subsystem as well as laid out the design for the Gemini launch vehicle.

Air Force Space Systems Division has published the Development Plan for the Gemini Launch Vehicle System. Using experience drawn from the Titan II and the Mercury development programs it is estimated the development of the launch vehicle will require a budget of $164.4 million. This includes a contingency fund of 50 percent to cover cost increases and unforeseen changes. ## Wednesday, March 21, 1962 McDonnell awards a$4.475 million subcontract to the Western Military Division of Motorola, Inc, of Scottsdale, Arizona. Western Military Division is to design and build the Digital Command System for the Gemini spacecraft. This is to receive in digital format commands from ground stations, to decode them, and to send the commands to the appropriate spacecraft systems. Two types of commands are anticipated: real-time commands for spacecraft functions, and stored program commands to update data on the spacecraft’s digital computer. The Digital Command System is to consist of a receiver/decoder package and three relay packages.

Air Force Space Systems Division awards a letter contract to the Aerojet-General Corporation of Azusa, California. This is to research, develop, and procure fifteen propulsion systems for the Gemini launch vehicle, and also for the design and development of related ground equipment. Aerojet was authorized to work on the engines on February 14th. Final engine delivery is scheduled by April 1965.

## Monday, March 19, 1962

McDonnell awards a $3.2 million subcontract to Advanced Technology Laboratories, Inc, of Mountain View, California. The subcontract is for the horizon sensor system for the Gemini spacecraft. One primary and one secondary horizon sensor are to be part of the guidance and control system. The sensors are to detect and track the gradient of infrared radiation between the Earth and outer space. McDonnell also awards a$400,000 subcontract to the Thiokol Chemical Corporation of Elkton, Maryland, for retrograde rockets. The solid-propellant retrorockets, four of which are designed to be put in the adapter section, are to start reentry or, in the event of a high-altitude suborbital abort, separate the spacecraft from the Titan II booster. It is believed that only slight modifications of a motor already in use are necessary, and that the qualification program will not need to be elaborate.

McDonnell awarded a $5.5 million subcontract to AiResearch. AiResearch is to provide the reactant supply for Gemini spacecraft fuel cells. The fuel cells, which are designed to provide power and water, are to store hydrogen and oxygen in two double-walled, vacuum-insulated, spherical containers in the Gemini spacecraft’s adaptor section, jettisoned at the start of the reentry procedure. ## Friday, March 16, 1962 The Air Force successfully launched a Titan II intercontinental ballistic missile at 18:09 Greenwich Time. This suborbital flight, the first full-scale test of the vehicle to also be Project Gemini’s booster, from Launch Complex 16, flew five thousand miles out over the Atlantic Ocean and reached an apogee of about eight hundred miles. This vehicle was serial number N-2. Launch Complex 16 has since its inauguration on December 12, 1959, been used for six launches of the Titan I, three of them successful. North American awarded a$225,000 subcontract to the Radioplane Division of Northrop Corporation today, as part of North American’s contract to design and develop emergency parachute recovery systems and test vehicles for the Paraglider Development Program.

McDonnell contracted with Vidya, Inc, of Palo Alto, California, today to test new ablation materials for the Gemini heat shields.

## Wednesday, March 14, 1962

The Gemini Project Office made a major decision about seat ejection. It is to be initiated manually, with both seats ejected simultaneously in case either ejection system is energized. The seat ejection is to be useful as a way to escape an emergency while on the launchpad, during the initial phase of powered flight (to an altitude of about 60,000 feet), or on reentry following a failure of the paraglider landing system.

The escape system is to include a hatch actuation system, opening the hatches before ejection; a rocket catapult to shoot both seats away from the spacecraft; and parachutes for the astronauts following their separation from the seat. The system is also to provide for survival equipment for the astronauts to use after landing.

The design is to allow for an automatic initiator in case this later becomes a requirement.

In other news the Manned Spacecraft Center issued its second analysis of the Gemini program schedule. This is the first to consider launch vehicles as well as the spacecraft. (The earlier analysis, of just Gemini operations, was published January 5.) Analysis of the Agena vehicles is limited, as their procurement began only with a request the Manned Spacecraft Center sent to Marshall Space Flight Center on January 31 for the eleven Atlas-Agena rendezvous targets believed needed.

The Gemini program is projected to use a number of test articles for engineering development, correcting a problem which had delayed the Mercury Program at times. The first, unmanned, qualification test is projected for late July or early August 1963. The second, manned, flight is now planned for late October or early November 1963. The first Agena flight is projected for late April or early May 1964. The remaining flights in the program are to be at roughly two-month intervals from then until the middle of 1965.

## Monday, March 12, 1962

Marshall Space Flight Center delivers to the Gemini Project Office a procurement schedule for Agena target vehicles. The Air Force Space Systems Division is to contract with Lockheed for 11 target vehicles. Space Systems Division is to put the Gemini Agena target vehicle program under the Ranger Launch Directorate.

Marshall expects that the delivery of a main engine qualified for multiple restarts will be in 50 weeks. This is an improvement in development time: the main engine is no longer considered the pacing item in the schedule for Agena development.

## Wednesday, March 7, 1962

The Gemini Project Office accepted McDonnell’s preliminary design for the Gemini main undercarriage for use in land landings.  It authorized McDonnell to proceed with the detail design.  Dynamic model testing of the undercarriage should begin around April 1.

McDonnell subcontracted to the Minneapolis-Honeywell Regulator Company of Minneapolis to provide the attitude control and maneuvering electronics system.  This is to provide the circuitry linking astronaut controls to the attitude and maneuvering controls and the reaction control system.  The contract is for $6.5 million. ## Monday, March 5, 1962 Westinghouse Electric Corporation of Baltimore received a$6.8 million subcontract from McDonnell. Westinghouse will provide the rendezvous radar and transponder system for the Gemini craft. The transponder is to be located in the Agena target vehicle.

Harold I Johnson, head of the Spacecraft Operations Branch (Flight Crew Operations Division of the Manned Spacecraft Center), circulated a memorandum on proposed training devices. The mission simulator should be capable of replicating a complete mission profile including sight, sound, and vibration cues, and be initially identical to the spacecraft, mission control, and remote site displays.

Training for launch and re-entry is to be provided by the centrifuge at the Naval Air Development Center (Johnsville, Pennsylvania), with a gondola set up to replicate the Gemini spacecraft interior. A static article is to serve as egress trainer. A boilerplate spacecraft with paraglider wing, used in a program including helicopter drops, will provide experience in landing on dry land. A docking trainer, fitted with actual hardware, capable of motion in six degrees of freedom, is to be used for docking operations training. And other trainers would be used for prepare for specific tasks.

The first regular business meeting between the Gemini Project Office and McDonnell occurred. Subsequent meetings are scheduled for the Monday, Tuesday, and Friday of each week. The initial coordination meetings had been held February 19, and introduction meetings were held the 19th, 21st, 23rd, 27th, and 28th.  The objective of these meetings is to discuss and settle differences in decision-making about the project.

## Sunday, March 4, 1962

Mercury Astronauts Scott Carpenter and Walter Schirra went through water-egress exercises, including practice with helicopter pickups.

## Friday, March 2, 1962

The Mercury 7 astronauts, who may be expected to continue on to Project Gemini, were today the guests of the United Nations, following yesterday’s “John Glenn Day” in New York City.  Glenn himself was spokesman during an informal reception given by Acting Secretary General U Thant.